Proof that an infinite product of discrete spaces may not be discrete

The product you describe is compact by Tychonoff's theorem, but the only compact discrete spaces are the finite ones.

The thought process behind this proof is strongly informed by the material in this blog post. A product of finite discrete spaces is an example of a profinite set or Stone space, and these behave in very particular ways.

Alternatively, the product you describe is second-countable, which an uncountable discrete space isn't.

More elementarily: The open sets in $\prod_{\mathbb N}\{0,1\}$ are precisely the (possibly infinite) unions of cylinder sets. Because even a single cylinder set is infinite, so is every nonempty open set. In particular, a singleton cannot be open, so the space is not discrete.