I'm trying to prove that $$D=\{(x,x):x\in[0,1]\}\subset [0,1]\times [0,1]\in \mathcal B_{[0,1]}\times\mathcal B_{[0,1]}$$
This problem is somewhat related to a problem in Folland, problem 2.46. I thought about taking sections of $D$, but not exactly sure how to proceed. I would love to get some help.
As observed in the comments it is enough to show that $\mathcal B ([0,1]\times [0,1])$ is contained in $(\mathcal B ([0,1])\times (\mathcal B [0,1])$. For this it is enough to show that any open set in $\mathbb R^{2}$ is contained in $(\mathcal B ([0,1])\times (\mathcal B [0,1])$. Since $\mathbb R^{2}$ is second countable any open set in it can be expressed as a countable union of sets of the type $U\times V$ where $U$ and $V$ are open in $\mathbb R$. Obviously, $U\times V$ is in $\mathcal B ([0,1]\times \mathcal B [0,1])$ so we are done.