Learn Hardcore French: Reste à la maison jusqu'à ce que le bus arrive.

Because it’s an imperative (a command). In French, positive commands drop the subject pronoun. You just use the verb in its imperative form (here reste), and the listener (tu) is understood.

Why is the imperative reste and not restes?

For -er verbs like rester, the tu form of the positive imperative drops the final -s. So tu restes (indicative) becomes reste (imperative). Exception: you keep the -s to maintain pronunciation before en or y (e.g. vas-y, parles-en).

What does jusqu’à ce que mean and when do you use it?

Jusqu’à ce que means until and introduces a subordinate clause expressing a time limit. It always triggers the subjunctive mood in the following verb.

Why is le bus arrive in the subjunctive mood, and how do we know?

After jusqu’à ce que, French requires the subjunctive. Here, arrive looks identical in form to the present indicative but is understood as subjunctive because of jusqu’à ce que.

Could we say jusqu’à le bus arrive instead?

No. Jusqu’à plus a noun works for simple expressions (jusqu’à midi, jusqu’à toi), but to link a full clause you need jusqu’à ce que + subjunctive.

Why use à la maison instead of chez toi or chez moi?

À la maison means at home in a neutral way. Chez toi (at your place) is more personal, and chez moi (at my place) would tell someone to stay at the speaker’s house, changing the meaning.

What’s the purpose of the apostrophe in jusqu’à?

French contracts jusque + à to jusqu’à for euphony, because jusque ends in a vowel and à begins with one.

Can I start with the time clause and say Jusqu’à ce que le bus arrive, reste à la maison?

Yes. You can front the subordinate clause. Just add a comma: Jusqu’à ce que le bus arrive, reste à la maison. The meaning remains the same.