Learn Hardcore Russian: Карта лежит на столе.

Because it’s the subject of the sentence. In Russian, the noun that performs the action or “is” in a state of being is in the nominative case. Here карта (map) is singular and feminine, so it appears as карта.

What does the verb лежит mean, and why is it used here instead of something like “is lying”?

Лежать means “to lie” (in the sense of “to be in a horizontal position”). Лежит is the 3rd person singular present tense of the imperfective verb лежать. It translates naturally into English as “lies” or “is lying.” Russian does not need a separate auxiliary verb “is” plus a participle; the single word лежит covers both.

Why is столе in the prepositional case?

When you talk about location “on” something with the preposition на, and you mean a static location (not movement toward or away), the noun that follows is put into the prepositional case. Here you’re describing where the map is located, so стол (table) becomes столе.

Could you use a different preposition instead of на?

You could say в + prepositional only if the object is inside something (e.g. в комнате “in the room”). A map can’t be “in the table,” so в столе wouldn’t make sense. For “on the table,” на is correct.