Learn Polish: Ilu obcokrajowców mówi po polsku? - How many foreigners speak Polish?

Because we are asking about obcokrajowców (foreigners), which is a masculine personal noun. Just like you are learning to use special numbers like dwóch or trzech for groups that include men, you must use the special "virile" form Ilu instead of Ile.

Why is the verb mówi (he/she/it speaks) in the singular, when we are asking about multiple foreigners?

This is a very important rule! When a sentence's subject uses a virile number (like dwóch) or a virile quantity word like ilu (how many), the verb must take the third-person singular form. Grammatically, the language treats the group as a single block, almost like asking "How much of the foreigners group speaks Polish?".

What case is the word obcokrajowców, and why?

It is in the genitive plural case. In Polish, words expressing quantity, including ilu (how many) and the virile numbers like dwóch (two), force the noun being counted into the genitive plural. The -ów ending is the standard genitive plural ending for masculine nouns.

I know the word for "Polish" is polski. Why does this sentence use po polsku?