Th reason Cid kidnapped Garnet was because he feared for her safety. He could smell a rat from all these dodgy dealings going on. It was the most wise move to remove Garnet at that time. Otherwise Brahne would have had direct and easy access to her Ediolons. Not to mention that Regent Cid promised Garnet's father as much. Regent Cid was an uncle to Garnet and it was only natural he did what he did. With communications between him and Brahne breaking off and the kingdom-kingdom atmosphere getting decidedly chilly, it was the only sensible thing he could do.


Regent Cid: "I once promised your father <gwok>, that should anything
happen, I would protect you. We've known about the
disturbances in Alexandria for some time. But had we acted
directly, it would've started a war. So, I asked Baku for
help. He and I go way back. The play was the perfect cover
to enter Alexandria and get you out. No one would suspect
Lindblum was behind it. We were <gwok> forced to take
action, because we knew Alexandria would never seek our
help. I'm relieved we were able to get you here."

Here's the quote.