Learn Danish: Jeg vil gerne rejse. - I would like to travel.

Because vil is a modal verb. Just like in English where we say "I must travel" instead of "I must to travel", Danish modal verbs are followed directly by the dictionary form of the next verb without using at.

I thought vil meant "will". Does it mean "would" here?

vil actually means "want to" in Danish, which is a very common trap for English speakers! By itself, jeg vil rejse means "I want to travel". To make it a polite "I would like to", we have to add the word gerne.

What exactly does gerne add to the sentence?

Literally, gerne means "gladly" or "willingly". When you combine it with vil (want to), the phrase vil gerne translates to "would like to". This is the most common and polite way to express a desire or request in Danish.

Why is it rejse and not rejser?

Because it follows the modal verb vil. Modal verbs are always followed by the bare infinitive (the dictionary form of the verb). In Danish, infinitives almost always end in -e, while the present tense usually adds an -r (rejser).