Learn Russian: Я хочу остаться здесь.

Because it follows the verb хочу (I want). Just like in English, when you combine a modal verb like хотеть (to want) with another action, the first verb changes to match the subject (I want = я хочу) and the second verb stays in its infinitive or dictionary form (остаться = to stay).

Is хочу an irregular verb form?

Yes! It comes from the verb хотеть (to want). It is a highly irregular verb in Russian. The stem changes depending on who is doing the wanting. For я (I), it becomes хочу. You will also see it change its vowels for plural subjects, like мы хотим (we want).

Why does остаться end in -ся instead of the usual -ть ending for verbs?

The -ся at the very end indicates it is a reflexive verb. While the standard infinitive ending is -ть, many verbs add -ся to the end. Sometimes this means an action is done to oneself, but for остаться (to stay), it simply makes the verb complete without taking a direct object. Notice that the soft sign (ь) stays right before the -ся.

How do you pronounce the -ться at the end of остаться?