Learn Hardcore Norwegian: Tom kjøper avisen i kiosken. - Tom buys the newspaper at the kiosk.

Kjøper is the present-tense form of the verb kjøpe (to buy). In Norwegian, you conjugate kjøpe in the present by adding -r to the stem:
jeg kjøper, du kjøper, han/hun kjøper, vi kjøper, dere kjøper, de kjøper.

Why is the object avisen in the definite form instead of en avis (a newspaper)?

When you mean “the newspaper” (a specific one), you use the definite form. In Norwegian you add the suffix -en to make a masculine noun definite:
avis → avisen (“the newspaper”)
By contrast, en avis means “a newspaper” (indefinite).

Why is there no article or preposition before avisen?

Avisen is the direct object of the verb kjøper. Norwegian does not require a preposition before a direct object; you put the object immediately after the verb in standard Subject-Verb-Object (SVO) order.

What does i kiosken mean, and why is i used here?

I normally means “in” or “at.” Here i kiosken literally means “in the kiosk,” but in English we’d say “at the kiosk.” The noun kiosken is definite (kiosk + -en = “the kiosk”), and i indicates location inside or at that place.