Invitation to visit a colleague's university, who usually provides support for this?
If you have been invited to visit, the default assumption in my realm is that the inviter will cover expenses unless they tell you up front that they don't/may not have money to fund all or any of your visit.

Usually the thing to do when you want to take an advantage of an open, informal invitation is contact the person, reminding them of your conversation, and ask about or suggest times to visit.  If their reply does not make it clear if they intend to fund your visit, just ask them.