Learn Hardcore Norwegian: Vi kjøper to lyspærer og skrur dem i lampen på verandaen. - We buy two light bulbs and screw them into the lamp on the veranda.

Because we’re talking about a specific lamp and a specific veranda (the lamp on the veranda), but just two bulbs in general. In Norwegian, definiteness is usually marked with a suffix:

lampe → lampen (the lamp)
veranda → verandaen (the veranda)
lyspære → lyspærer is plural indefinite (two bulbs); the definite plural would be lyspærene (the bulbs).

What does dem refer to, and why is it dem and not something else?

Dem refers back to to lyspærer and is the object pronoun for plural “them.” Subject “they” is de. Singular objects would be den (for masculine/feminine nouns) or det (for neuter nouns).

Is it okay to write de instead of dem here?

Not in standard written Bokmål. Use de for subjects and dem for objects. In informal speech, many people say dem as a subject too, but avoid that in writing.

What does skru … i mean, and how does it differ from things like skru på/av/fast/inn?