Learn Swedish: När hon är ledsen stänger hon sig inne. - When she is sad, she shuts herself in.

This is the core rule for reflexive particle verbs in Swedish. The reflexive pronoun (sig) must always be placed directly after the verb and subject, and before the particle (inne). You cannot put the particle before the reflexive pronoun.

Why does the second part of the sentence say stänger hon instead of hon stänger?

This is the V2 (verb-second) rule in action. The entire subordinate clause När hon är ledsen counts as the first block of information. The verb stänger must come in the second position, which forces the subject hon to swap places and come immediately after it.

Why does this verb use inne instead of in?

In Swedish, in usually shows movement towards the inside, while inne shows a fixed location or a state of being inside. The reflexive particle verb stänga sig inne focuses on the resulting state of keeping oneself isolated or contained, rather than the physical movement of going inside a room.

What would it mean if I used henne instead of sig here?

If you wrote stänger hon henne inne, it would mean she is locking a completely different woman inside. You must use the reflexive pronoun sig to show that the subject (hon) is performing the action on herself.