Learn Russian: В этом кафе так много людей, что нам некуда повесить пальто. - There are so many people in this cafe that we have nowhere to hang the coat.

негде describes a static location (answering где, "where at"). However, the verb повесить (to hang) involves moving an object to a destination. In Russian, this requires a directional word (answering куда, "where to"). Therefore, the correct predicative adverb for "nowhere to" is некуда.

Why is the pronoun "we" translated as нам (dative case) instead of мы?

This is the core structure for predicative negative adverbs like некуда, негде, and некогда. They form impersonal sentences where the logical subject (the person experiencing the lack of a place or time) must be in the dative case. It translates literally to something like "there is nowhere for us to hang..."

Why is the word for "people" людей instead of the nominative люди?

The phrase uses много (many / a lot). After words that express a quantity, the noun that follows must take the genitive case. людей is the genitive plural form of люди.

Since "coat" is the direct object being hung up, why doesn't пальто change its ending?

пальто is a borrowed foreign word and belongs to a special group of Russian nouns that never change their endings. They are "indeclinable." Even though it is the direct object here (which would normally require the accusative case), it remains exactly the same: пальто.