I often see reference to battles and armies numbering in the tens to hundreds of thousands of troops. How do we know the accuracy or validity of those sizes?

It can often depend on who's giving the numbers, and also whose side's numbers they're giving. More can always be said on the matter, if anyone would like to address the matter of army sizes, please go ahead and post up! More insight is always welcome even when there are previous posts.

For the meantime, OP, here's some older posts on how far we can trust the given numbers:

u/Iphikrates speaks as to why we trust Greek numbers versus why Persian numbers seem off;
u/Hergrim looks at numbers for the Battle of Crecy;
and u/Goiyon's first post shows how we can verify some given numbers.