Learn Finnish: Opettaja aloittaa tapaamisen. - The teacher starts the meeting.

Because it is the object of the sentence. In Finnish, when a subject does an action to a whole, complete thing (like starting the entire meeting), the object takes an -n ending. The basic dictionary word is tapaaminen.

I have learned that alkaa means to start. Why are we using aloittaa in this sentence?

Aloittaa is used when a person actively starts something else. Because the teacher is starting the meeting, we use aloittaa. You use alkaa when something starts by itself, for example: Tapaaminen alkaa (The meeting starts).

How does the word tapaaminen relate to the verb for meeting someone?

It comes directly from the verb tavata (to meet). Adding -minen to a verb stem is a common way to turn it into a noun, similar to adding -ing in English. So, tavata becomes tapaaminen (a meeting). Notice the spelling change where the v becomes a p!

Since this lesson is about telling time, how would I say that the teacher starts the meeting at nine o'clock?

You can just add the time expression to the end of the sentence. You would say: Opettaja aloittaa tapaamisen kello yhdeksän (at nine o'clock) or Opettaja aloittaa tapaamisen yhdeksältä (at nine).