Learn Hardcore Russian: После совещания мне надо позвонить начальнице. - After the meeting, I need to call my boss.

Because the preposition после (after) requires the genitive case.
совещание (a meeting) is neuter nominative/accusative; its genitive singular form is совещания.

How do I know that совещания is genitive singular and not something else?

For neuter nouns ending in -ие / -ание, the genitive singular typically ends in -я:
совещание → совещания.
Also, после + genitive is a very common fixed pattern for after X.

What does мне mean here, and why is it not я?

мне is dative of я and literally means to me. In this structure, Russian often marks the person who has the obligation/need in the dative:

мне надо + infinitive = I need to / I have to (do something) (literally: to me it is necessary).

Is there a subject in this sentence? Who is doing the calling?