Why is force a localized vector?

When you touch an object, the charged particles (electrons and protons) in your hand exert an electrostatic repulsive force on the charged particles in the object. This force is predominantly exerted on the atoms and molecules on the surface of the object, causing them to be displaced slightly from their previous positions.

This displacement deforms the chemical bonds within the solid, causing the bonds to exert a force on the surface molecules' neighbors. These neighbors are displaced, which causes them to exert a force on their neighbors, and this continues on and on. This is the mechanism by which force is "transmitted" through solid objects (though that term is not really an accurate description of what's happening).

In physics a free vector is defined as having a direction and a magnitude; location does not matter. See Symon, Mechanics. This definition is necessary to consider the transformation of the components of a vector between different coordinate systems, one coordinate system in translation and rotation with respect to the other.

Of course the location of a vector does determine its effect; for example, the force vector has to be localized at a particle to act on the particle. Some developments (mostly mechanical engineering texts) go to great lengths to discuss bound (or localized) and sliding vectors. I prefer how Davis approaches this in his book Introduction to Vector Analysis, where he explains that physicists regard a vector as a free vector, recognizing nevertheless that the effect of the vector may depend on where is it applied.

Regarding torque as a vector, the torque vector explicitly depends on the origin about where the torque is calculated; using a different origin produces a different torque. Torque addresses the rotation with respect to the origin, produced by a force acting at a specific point, either at a single particle, at a specific position on a rigid body, or at a specific particle in a system of particles. A force effectively acting at the center of mass (such as gravity) does produce a torque if the origin is taken as some point other than the center of mass where the cross product of distance and force is not zero; if the center of mass is taken as the origin this force produces no torque. Just as the external forces depend on the definition of our system, the torque depends on the origin used. For a body not constrained to have a specific point fixed, the torque is typically taken about the center of mass as that simplifies the evaluation. Conversely, if a point is fixed that point is taken as the origin for evaluating torque. See any basic physics text such as one by Halliday and Resnick.

The mechanics relationships using vectors for force, torque, etc. were developed and shown to be consistent with Newton's laws of motion which in turn agree with experimental observations. Actually, vectors were first used for electrodynamics then later applied for classical mechanics. The concept is that a force acts at a point and affects the motion depending on its magnitude and direction regardless of the specific microscopic characteristics of the force, whether it be electrodynamic or gravity (strong and weak nuclear forces are not addressed in classical mechanics and electrodynamics). Other concepts besides vectors, such as quaternions, can also be used to evaluate the motion.

Bottom line: vectors are a mathematical construct that explain the dynamics of motion, regardless of the specific microscopic phenomena involved.