Learn Hardcore Russian: Собрание начинается через десять минут, поэтому мне нужно выйти сейчас. - The meeting starts in ten minutes, so I need to leave now.

Собрание is the grammatical subject of the sentence, so it’s in the nominative case: Собрание начинается…
The noun собрание is neuter because it ends in -е / -ие (a common neuter pattern in Russian). That’s why the verb is начинается (3rd person singular) with no gender marking in the present tense.

What does начинается mean grammatically, and why is it reflexive (-ся)?

Начинается is the present-tense form of начинаться (imperfective).
The -ся makes it a “self-happening” / passive-like construction: the meeting is starting / begins (i.e., it starts by itself, not “someone starts it”). Compare:

Он начинает собрание = He starts the meeting (active, someone is doing it).
Собрание начинается = The meeting begins (focus on the event starting).

Why is present tense used (начинается) if it’s about the near future?

Russian often uses the present tense of an imperfective verb to talk about scheduled or imminent future events, especially with a time expression:
Собрание начинается через десять минут = The meeting starts in ten minutes.
You can also see начнётся (perfective future) in similar contexts; it can sound a bit more like a single definite start.