Learn Finnish: Opiskelijan täytyy harjoitella paljon.

This is because of the word täytyy (must). In Finnish, when you express obligation, the person who has to do the action takes the genitive case, which usually means adding an -n to the end. So, "the student must" becomes opiskelijan täytyy.

If I wanted to say "I must practice a lot", how would this sentence change?

You only need to change the subject to the genitive form of "I". The word for "I" is minä, and its genitive form is minun. The rest of the sentence stays exactly the same! So you would say: Minun täytyy harjoitella paljon.

Does the verb täytyy change its ending depending on who is doing the action?

No, it doesn't! Normal verbs change their endings to match the subject (like minä puhun, hän puhuu), but täytyy is a special word that is frozen. It always stays exactly the same, whether it is "I must" (minun täytyy) or "the student must" (opiskelijan täytyy).

Why is the second verb, harjoitella, in its basic dictionary form?

When you have a chain of verbs with täytyy, the action being done always stays in the basic infinitive form. It is exactly like English, where we say "must practice" instead of "must practices".