What does the Cayley graph of the Grigorchuk group 'look like'?

I do not know about embedding in higher dimensional spaces, but here is a general theorem about dimension 2:

Suppose that $G$ is a finitely-generated group which admits a Cayley graph that has an accumulation-free (i.e., proper) topological embedding in the plane. Then $G$ cannot have intermediate growth.

This is a corollary of Theorem 1.1 here, since groups acting properly discontinuously on planar surfaces are well-understood (see references in the link above) and, in particular, they cannot have intermediate growth. Now, if you have an isogonal embedding in the hyperbolic plane $H^2$, then this embedding is accumulation-free in $H^2$. Applying the inverse of the exponential map for $H^2$ to such an embedding we obtain an accumulation-free embedding in $R^2$.

I am not sure what happens if you allow non-proper embeddings, take a look at the references in the link, maybe you can find further information about this.

One more thing: If you drop the isogonality requirement, then every countable graph will properly embed in 3-dimensional space (hyperbolic or Euclidean does not matter).

Addendum: If a planar Cayley graph does not admit a proper planar embedding, then it is easy to show (see the same link above) that the group has at least 2 ends, i.e., it cannot have intermediate growth as well.

It's this last reduced graph that I'm particularly curious about: is it planar?

An excellent question! I've been trying to find an answer to it, but haven't so far. To my knowledge, it's open.

Here's what $B_{10}$ in the Cayley graph of Grigorchuk group looks like when the $abcd$ "kites" are collapsed and leaves removed:

Here, $B_{10}$ is the ball of words of length 10 or less around identity, with the standard generating set. Each node represents the $K_4$, and each edge corresponds to $a$.

The graph was laid out with Graphviz (neato engine), as is the next one, for $B_{11}$:

I haven't checked whether the larger graphs are planar, but I'd conjecture that they are.

I also didn't experiment enough with edge lengths; these two embeddings are quasi-isometric due to how layout engine works when unconstrained

Certainly a very interesting question, and I didn't yet give up hope of exploring it deeper some day :)