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

If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and makes a new Hamiltonian. Hence a fixed point in an RG flow means it's scale invariant, and every textbook says therefore it's a critical point where a phase transition will occur.

But why? It seems scale invariance (meaning correlation length diverges) is considered as a feature of a system in a critical state, but I can understand neither why the correlation length diverges nor why the system is scale invariant at the critical point.

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.
The maths then usually shows you that the correlation length $$\xi$$ goes as $$\propto (T-T_{\mathrm{c}})^{-\nu}$$, that is $$\xi\rightarrow\infty$$ as $$T\rightarrow T_{\mathrm{c}}$$. From which scale invariance follows.