I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is a result I hadn't seen before. I'm struggling to see the connection.
So far I have shown that $\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G] \cong \mathbb{F}_p[G]$. How can I find the dimension of $\mathbb{F}_p[G]$, and relate that to the $p$-rank of $G$?
As Nate suggested in a comment, the correct formula is $r_p(G)=\mathrm{dim}_{\mathbb F_p}(\mathbb F_p\otimes_{\mathbb Z} G)$ (where $r_p(-)$ denotes the $p$-rank of the argument). To prove this formula you can proceed as follows: Note first that $\mathbb F_p\otimes_{\mathbb Z} G=\mathbb Z/p\mathbb Z\otimes_{\mathbb Z} G\cong G/pG$. Moreover, for any Abelian group $G$ which is a direct sum of cyclic groups (so, in particular, for any finite Abelian group $G$ as in your question), the following isomorphism holds for all $m>0$: $$ G/mG\cong G[m]:=\{g\in G:mg=0\} $$ (this is a nice exercise). When $m=p$ is your prime, $G[p]$ is just the $p$-socle of $G$ and it is easy to see that $r_p(G)=\log_p|G[p]|$ (where $|-|$ denotes cardinality). Finally, by the above isomorphisms, $$ \log_p|G[p]|=\log_p|G/pG|=\log_p|\mathbb F_p\otimes_{\mathbb Z} G|=\mathrm{dim}_{\mathbb F_p}(\mathbb F_p\otimes_{\mathbb Z} G), $$ where the last equality comes from the fact that, for any finite dimensional vector space $V$ over a finite field $\mathbf k$, one clearly has that $|V|=|\mathbf k|^{\dim_{\mathbf k}(V)}$ and, therefore, $\dim_{\mathbf k}(V)=\log_{|\mathbf {k}|}|V|$.
To see that the formula in your original question cannot be right, just observe that, for any group $G$ (Abelian or not), the underlying Abelian group of the group algebra $\mathbb Z[G]$ is just a free Abelian group of rank $|G|$, that is, as Abelian groups, $\mathbb Z[G]\cong \mathbb Z^{(G)}$. In particular, $\mathbb F_p\otimes_{\mathbb Z}\mathbb Z[G]$ is an $\mathbb F_p$-vector space of dimension $|G|$ (independently on your choice of $p$). Hence, the dimension of this $\mathbb F_p$-vector space cannot be computing the $p$-rank of your finite Abelian group $G$ (as this value has actually no dependence on $p$).