What is game sharing and how does it work?

An account can have a single Xbox set as their home console. Anyone on that console has access to that account's digital games and subscriptions.

If I were to add a friend's account to my Xbox and set it as a home account, would they have access to my personal data?

I'm assuming you mean if you set your console as their account's home. If so, no, they will not gain access to your data.

Can I download their games onto my Xbox and play them without having to be signed into their account?

If your console is their account's home then yes

Can I just download their games and then sign out of their account and remove it as a home account?

If your console stopped being their home console you'd lose access to their digital games unless they were signed into your console