Why does the Ising model at the critical point have scale invariance?

I don't think there is an accepted answer to the why this happens. This is usually referred to as the 'scaling hypothesis', i.e. that in the vicinity of (continuous) phase transitions, thermodynamic quantities and correlation functions typically behave as power laws characterized by universal exponents, which are independent of microscopic parameters of a system.

First things first, scale invariance and correlation length ($xi$) divergence go hand in hand. The correlation length basically sets the lengthscale for the physical phenomenon of interest: if I wiggle a particle at position $x$, this effect will be felt up to a distance $x+xi$. Is the system is scale invariant, meaning the same phenomenon is present at short, intermediate, and long distances with the same intensity, then $xi$ cannot be finite. Hence it must be infinite.

It should also be noted that, realistically, you don't "really" have scale invariance on all scales. I mean, if you zoom in enough, you'll get to subatomic structures which obviously do not take part in phase transitions like liquid-gas or magnetisations. Hence why the visual representations of the RG method show zooming out rather than zooming in.

A possible answer to the why question is the following.
A phase transition is characterised by a non-analytic free energy. That is, something blows up and goes to infinity at the critical point. Infinity is infinity, there are no nuances of infinity. So, close enough to the phase transition so as to be dominated by this infinity, the specifics of the material and of the scale at which we are looking become irrelevant. So you would expect to approach a "universal" behaviour across different materials, different configurations, and different lengthscales for that matter.
The maths then usually shows you that the correlation length $xi$ goes as $propto (T-T_{mathrm{c}})^{-nu}$, that is $xirightarrowinfty$ as $Trightarrow T_{mathrm{c}}$. From which scale invariance follows.