Learn Turkish: Salataya yeşil biber koydum. - I put green pepper in the salad.

In English, we say we put things "in" or "on" something, which sounds like a static location. But in Turkish, the verb koymak (to put) focuses on the movement towards a destination. Because of this, the destination takes the dative case ending -(y)a or -(y)e (meaning "to" or "into"). You are literally saying "I put green pepper into the salad."

Why isn't the accusative case used here (like yeşil biberi)?

You only use the accusative case when the direct object is specific or definite, like if you were saying "I put THE green pepper in the salad." Because you are talking about green pepper in a general, indefinite sense, it stays in its plain dictionary form.

If I put several green peppers in the salad, should I say yeşil biberler?

Usually, no. In Turkish, when an object is general and indefinite (and placed right before the verb), it is kept in the singular form even if the meaning might be plural. Saying biberler sounds a bit unnatural here unless you are referring to specific peppers, which would then also need the accusative case (yeşil biberleri koydum).

The English word "put" is the same in present and past tense. How do we know this Turkish sentence is in the past tense?