If you are a professional horse dentist - and my girlfriend tells me that's a real job - then you can tell a lot about the health of a horse from looking at its teeth.
That's something you should do if you want to BUY a horse. It's like checking a car's engine before you hand over the money.
But if someone gives you a car for your birthday you shouldn't go checking in the engine saying things like 'Ugh it's a bit low on oil' or 'this fan belt has seen better days'. Just say 'thank you for the free car that's very kind of you'.
So 'looking a gift horse in the mouth' means 'being ungrateful'.
If your partner proposes an option that isn't your first choice you could use this phrase because getting something is better than nothing.
- "I think we should use the money to buy new computer equipment for the library."
- "Well, most people have computers at home these days so I don't think that's the best use of the cash. But I wouldn't look a gift horse in the mouth - the library is near my house so those computers would be handy if my laptop ever broke."
