I'm trying to sign some MiL FAs and I'm getting the same message from most of them: up your offer. When I go to increase my offer, I get an error message saying "A minor league contract will default to 1 year for the player's minimum required salary. No bonuses are acceptable.".

How can other teams be offering more money? Am I missing something?

They are either offering a minor league contract with an invitation to spring training, or they are offering a big league contract.

I'm pretty sure that no one is offering these scrubs big league contracts. But, if I up my offer to minor league with invite, how will they decide "who's higher" if we're both offering the same thing?

I sign a lot of minor-leaguers early on.  If another team (in addition to mine) has offered them the same MiL+Spring Invite, and has the winning bid, I'll get a message like this:

"While my client appreciates your offer, another organization has made a similar one. After reviewing the franchises' depth charts, I believe my client has a better shot of reaching the bigs with the other club. If you really want him, you're going to have to increase your offer. We'll give you a little time to decide."

I don't know if there are other tie-breakers.