Bloodboil fever reads:

"Attach to a target unit.Action: At the beginning of its controller's turn, put a resource token on this card. Then, deal X damage to the attached unit. X is the number of resource tokens on this card."

If the controller of the unit removes the attachment through other card effects, do the resource token remain on the card or are they removed when the attachment is removed?

the token are on the discarded attachment so they are removed

How do you use that card anyway?

I think you would like to attach this support card to enemy unit, but if you are able to play support cards only to your capital then i don't know how that card works.

Attachments are an exception. You can attach them to other players' cards, as long as the text of the Attachment card doesn't limit it to your cards.