DM me if you need it explained any simpler - I have put the main point you haven't understood in bold, but basically:
For every item in the auction house the game has a "current highest bid" which is visible to everyone, and a "current highest bidder has said they would pay up to a value of Y" which is hidden. If nobody has made a bid on that item it is the same as the listing price.
When you are making a bid on something, you are not saying "I will raise that price to my new bid" you are saying "I will pay up to my new bid for that item, but get me the cheapest price up to that. Here is the money up front to cover it and if there is any change left give it back at the end of the auction"
The game automatically checks that hidden "current highest bidder would pay up to Y" and compares your bid to it.
Is your bid higher than the hidden bid?
No?
OK, they had already said they would pay more, your highest bid wasn't enough but the new "cheapest price" that they can get it for is $1 higher than yours, so they REMAIN the highest bidder, but they now have to pay $1 higher than you were willing to pay.
It now shows that new price as the "current bid", and still has their hidden "but they would pay up to this" behind the scenes.
Yes?
Congratulations, you were willing to pay more than them - the new highest bid will be $1 higher than whatever their hidden maximum would be.
You are now the highest bidder, and that hidden value is whatever you had said you would pay up to.
The auction house message the previous highest bidder with a "sorry, but TakedaVigor was willing to pay more, here is a refund of the money you left with us to bid on your behalf"
If you are STILL the highest bidder at the end of the auction, and it ended up selling for less than you had said you would be willing to pay, you get given the item AND your surplus bid money.