How one uses the definition of observers in General Relativity?

This is a very interesting question. You are right in general relativity, observers cannot extract measurable quantities from a test particle or compare frame dependent information with another observer unless they meet at the same point or come close enough so that the spacetime can be considered, effectively, as flat and then a SR like situation is recovered. In general relativity, the Lorentz transformation between two frames is only possible locally. This is quite a different situation than in SR. And indeed this limits the role of observer notion in general relativity and cosmology. That is the reason why in general relativity the meaningful physical quantities are those which are observer independent, like, the line element, proper time and the other tensorial quantities (the metric tensor $bf{g}$, electromagnetic strength tensor $bf{F}$, etc.).

Nevertheless, this is quite a realistic situation; because real life observers can only measure physical quantities locally. The measurable quantites such as electric field strength, magnetic field strength, and in general energy-momentum tensor are local quantities. The EFE, written in the following form (in the component form), $G_{ab}(x)=-kappa T_{ab}(x)$, is only meaningful in a chart $(U,x)$; and a chart only provides information about localised region $(U)$ in the manifold.

Why does the observer in SR differ from the observer in GR?

The answer is a purely mathematical one: because the manifolds, in general, do not have vector space structure. The position vector and hence the coordinates only make sense in a vector space, but in a manifold the position vector loses its meaning. In a general manifold, to talk about coordinates we need to focus on a localised region of the manifold, the chart, isomorphic to $R^d$ ($d=$ dimension of the manifold). Now, SR considers the spacetime to be Minkowskian, which is, interestingly enough, isomorphic to $R^d$, globally or as a whole, and thus have both manifold and vector space structure. Therefore, observers in SR can set up coordinates (or reference frame(s)) which can span the whole spacetime. As a result the Lorentz transformation between the reference frames set up by two obsevers is valid all over the spacetime. But in GR, in presence of gravity, the spacetime is a curved manifold. Thus observes cannot set up reference frames which explore the whole spacetime, making the role of the observer extremely local, as you said. As @Emil pointed out in the comment, one observer can definitely parallel transport its tangent space (the observers' reference frame), to the location of other observer and then facilitate the Lorentz transforamtion or any other, but this does not help the situation, as the two observers have to meet in the same point (that's why the transportation!). Thus observers in GR can measure and tally only local observations.

Observers' measurements are local, but this does not imply that global inferences are impossible

The key here is symmetry. If the quantities which we are interested in, follow a pattern then the whole spacetime needs not be explored, a study over a local region can be extrapolated to figure out the global structure of the spacetime. For example, in cosmology the existence of six space-like killing vector field is assumed, which essentially implies, the matter distribution in the universe is homogeneous and isotropic in space, of course in large scale. This is the simplest kind of matter distribution possible. Due to the symmetry in the metric, the universe has constant spatial curvature everywhere. Now any local measurement of the spatial curvature reveals the global geometry of the spacetime. From the information of metric tensor, one can also find out, if possible, the conformal space transformation and study a great deal about the global properties of spacetime.

How to use observers in GR

The approach of GR in which observer frames are used is known as tetrad formalism or Cartan formalism. At each point of the curved manifold, it is possible to construct frames (vierbein), consisting of four orthonormal set of vectors {$e^alpha_a$} (one time-like and three space-like vectors), i.e. it is possible to construct a frame bundle on the manifold. Now, an observer is, precisely, a smooth section in the frame bundle. A section in the frame bundle is an integral curve of the time-like vector ($e^alpha_0$) field. And the three spatial vectors attached with the time-like vectors forms a spatial triad {$e^alpha_1,e^alpha_2,e^alpha_3$}. Here the greek indices denote the chart coordinates in the manifold and roman indices denote the local frame coordinates. In the local frame the metric is ordinary Minkowskian metric (local flatness). The relation between manifold's metric and the frame metric is as follows, $$e^alpha_ae^beta_bg_{alphabeta}=eta_{ab}$$ And, $$e^a_alpha e^b_beta eta_{ab}=g_{alpha beta}.$$ Thus, any tensorial quantity can be defined in the frames or coframes (sections in the co-tangent or co-frame bundle), for example, metric tensor can be expressed in the coframe as, $$g=-sigma ^{0}otimes sigma ^{0}+sum _{{i=1}}^{3}sigma ^{i}otimes sigma ^{i},$$ where for Schwarzschild vacuum the sigmas are, $$ sigma^0 = -sqrt{1-2m/r} , dt, ; sigma^1 = frac{dr}{sqrt{1-2m/r}}, ; sigma^2 = r dtheta, ; sigma^3 = r sin(theta) dphi.$$ It is convenient to change the basis from {$e^alpha_a$} to {$l^alpha,n^alpha,m^alpha,bar m^alpha$}, defined as, $$l^alpha=frac{1}{sqrt{2}}(e^alpha_0+e^alpha_1),$$ $$n^alpha=frac{1}{sqrt{2}}(e^alpha_0-e^alpha_1),$$ $$m^alpha=frac{1}{sqrt{2}}(e^alpha_2+ie^alpha_3),$$ and $bar m^alpha$ is the complex conjugate of $m^alpha$. As these vectors follow the following properties, $$l^alpha l_alpha=n^alpha n_alpha=m^alpha m_alpha=bar m^alpha bar m_alpha=0,$$ the set {$l^alpha,n^alpha,m^alpha,bar m^alpha$} is called the null tetrad. As the null tetrad is related to the metric tensor in the following way, $$g_{mu nu}=l_mu n_nu+l_nu n_mu-m_mu bar m_nu-m_nu bar m_mu,$$ one can use the null tetrad to compute various tensorial quantities in the local frames and then convert it back on the manifold. For an example, E.T. Newman and A. I. Janis (J. Math. Phys., 6, 915, 1965) used this technique to produce a simple derivation of the Kerr metric.

I believe the best way to sum up the idea of observers used in almost all treatments of standard Special Relativity is what Schutz says in his General Relativity book: [...]

I can accept that this is a correct quantitative description of presently available treatments. But I cannot accept the characterization of observers as it is suggested by Schutz (and others), because it appears to deny what to me is an indispensible element of discourse of (the geometric-kinematic part of) Einstein's theory of relativity, and Special Relativity in particular.

Namely, to consider individual identifiable observers attached to (or even identified as) individual identifiable material points, as repeatedly and consistently described by Einstein himself (e.g. here and here):

• each capable of observing and identifying and recognizing others, and in turn of being observed and recognized;

• each capable of preserving the collected observations in memory and of determining which observations he, she, or it had collected in coincidence, or in which order;

at least in principle, for the purpose of thought-experimental description and understanding; and more or less even in practice.

On the other hand, in General Relativity things are different.

Apparently as represented by Schutz et al.; but certainly not for the notion of observer as individual capable of collecting and ordering observations.

[...] together with an orthonormal basis

Which foremost raises the question how an individual observer ought to determine such a basis in the first place.

[One observer] cannot describe worldline of particles or other observers, nor anything of the sort. [...]

Surely any one observer can (be thought of) observing and recognizing others having observed his/her/its own (signal) indications; and consequently, signal by signal, determine whose corresponding ping echos had been received back in coincidence, or in which order, or "not yet at all". This capability is ascribed to observers already in Einstein's initial presentation of SR, 1905.

From the interelations between such determinations of individual observers follow descriptions of their collective geometric relations between each other; such as "ping-coincidence lattices" described here.

[...] what gives meaning to the coordinates.

The possible sprinkling of events (or likewise: of selected observers, and their individual ordered sets of indications) with coordinate tuples, for the purpose of representing the geometric relations between events (or likewise: for representing the frame relations between the selected observers) through the "natural" topological or even metrical properties of real number tuples, is of course only subsequent and secondary to the determination of the geometric relations under consideration.

VacuuM gave an excellent answer but I will say some more words to clarify some ideas.

It is a postulate of physics that inertial observers should measures the same thing that is, if an inertial observer $A$ set up a experiment and an observer $B$ set up an identical experiment than their result should be the same. In special relativity we have global inertial observers while in general relativity we have local inertial observer. Let us se why.

Suppose in special relativity (flat space ) an observer $A$ throw a ball with velocity $v$ to wall, that is at distance $d$ in relation to this observer. If this observer receives the ball with velocity $v'$ than another observer $B$ who makes the same experiment in his rest frame will receive the ball with same velocity $v'$. Now first in curved spacetime we can not have a non local static wall in relation to an observer. I am giving this example because of my lack of imagination. But assuming we could have a static wall in relation to an observer if this two observer make the same experience than the velocity that they will receive the ball will be different. Due to curvature of spacetime if the observer $A$ throw a ball with velocity $v$ it will arrive at the wall with velocity $v_A$ and if the observer $B$ throw the ball with the same velocity $v$ it will arrive at the wall with a different velocity $V_B$.Further if the observer $A$ makes the same experience in a different time it will have a different result. So we have to stay locally to have coincident result between inertial observers