Learn Russian: Мужские брюки здесь.

This is because of a very common Russian spelling rule. The base of the word ends in к. In Russian, you can never write the letter ы after к, г, х, or the hissing letters (like ш and ж). Because of this rule, the standard plural adjective ending -ые must change to -ие.

It translates to 'men's', so is мужские a possessive pronoun or an adjective?

Even though it translates to the possessive 'men's' in English, мужские is a regular adjective in Russian, literally meaning 'male'. Since it describes the plural noun брюки, it takes a plural adjective ending rather than acting like a possessive pronoun (such as 'my' or 'his').

Does the word брюки (trousers) have a singular form?

No. Just like the words 'trousers' or 'pants' in English, брюки only exists in the plural form in Russian. This means any adjectives describing it must always be plural as well.

Why is there no word for 'are' in this sentence?

In Russian, the verb 'to be' (am, is, are) is almost always skipped in the present tense. You just put the subject and its location together, so the sentence literally reads as 'Male trousers here'.