Learn Polish: Dlaczego obiecujesz wszystko obcym ludziom? - Why do you promise everything to strangers?

The verb obiecywać (to promise) requires the Dative case for the person receiving the promise. Think of it as promising something to someone. In Polish, that "to" direction is handled by putting "strangers" (obcy ludzie) into the Dative case.

How do you form the Dative plural for nouns like ludziom?

ludziom comes from the plural noun ludzie (people). The Dative plural is actually one of the easiest cases to learn because it is incredibly regular: almost every single noun, regardless of gender, simply takes the -om ending!

Why does the adjective obcym end in -ym?

Adjectives in the Dative plural also have a very regular pattern: they end in -ym or -im, completely ignoring noun gender. Since obcy (strange or foreign) has a hard stem, it takes the -ym ending to match ludziom.

What case is wszystko in this sentence?

It is in the Accusative case because it is the direct object (the thing being promised). Fortunately, the Accusative form of wszystko (everything) looks exactly the same as its basic dictionary form, so you do not have to change anything!