Learn Danish: Jeg vil gerne sidde herinde. - I would like to sit in here.

Because the verb sidde (to sit) describes a stationary position, not movement towards a destination. In Danish, spatial adverbs have two forms: one for staying in a place (ending in -e, like inde) and one for moving to a place (without the -e, like ind). Since you are just sitting and not moving into the room, you use the static form herinde.

What is the difference between saying her and herinde in this sentence?

While her simply means here, adding -inde adds the specific meaning of inside. By saying herinde, the speaker is emphasizing that they want to sit in here (for example, inside a café or building) as opposed to out there (derude).

Why is gerne included with vil in this sentence?

Vil on its own means want, which can sound very direct or demanding if used for a request. Adding the adverb gerne (gladly) softens the phrase vil gerne to mean would like to. This is the standard, polite way to express a preference in Danish.

Why is sidde in its infinitive form without the word at in front of it?