Learn Hardcore German: Ich brate die Paprika kurz in Öl. - I briefly fry the bell pepper in oil.

anbraten: to brown/sear briefly at the start → Ich brate die Paprika kurz in Öl an. (very common in recipes)
rösten: to roast/toast; often drier heat or stronger browning, sometimes without much oil
schmoren: to braise/stew slowly with liquid (not “briefly”) So braten or anbraten fits best with kurz .

Paprika is feminine in German when it means the vegetable (bell pepper), so you say die Paprika (singular) and die Paprikas or often die Paprika (plural, depending on context/region).
Note: der Paprika can appear when referring to the spice (paprika powder), though usage varies. In this sentence (frying), it’s clearly the vegetable → die Paprika.

What exactly does brate mean here, and why is it brate (not braten)?

braten is the infinitive (to fry / to sauté / to pan-fry).
brate is the 1st person singular present tense: ich brate = I fry / I’m frying.
Conjugation: ich brate, du brätst, er/sie/es brät, wir braten, ihr bratet, sie/Sie braten.

Why is it die Paprika (accusative) rather than dative?

Because die Paprika is the direct object of braten (what you are frying). Many action verbs take an accusative object.
You can see the case from the article: feminine die stays die in both nominative and accusative, so the article doesn’t change—but the role is still accusative (direct object).