Provably fair means you can prove that a casino did not alter the result of a bet to deliberately make you lose.
It was originally developed by Bitzino IIRC and the industry standard provably fair system used on dice sites (like PRC) was invented by dooglus of Just-Dice.
Any dice site not using this system, there are quite a few, should not be trusted at all.
This system means it is impossible for site operators to alter your bet to make you lose without the chance of them being caught.
Note that this can still happen, look at the old dicebitco.in scam, but this system allowed them to be caught.
There are a lot of people who misunderstand this. Provably fair does not mean that you will win or not lose money.
It does not mean an operator cannot cheat investors if they allow an invest option.
It does not mean that you will definitely be paid your winnings either.
Those are all separate trust issues that you should take into account based on the community, reviews and by observing the sites you wish to play at.
As for FIAT sites using a probably fair system? It doesn't look likely any time soon.
I went to the iGaming conference and met with large operators and game providers and from what I seen there they probably wouldn't want this.
Also it seems 99% (probably more) of players don't care about it, even in our Bitcoin gambling community it seems most don't care and even the ones that do do not understand what it means, how it works or how to verify it.
I would recommend reading this tutorial/help page on what provably fair is and how it works by NlNico
http://dicesites.com/provably-fairBut again it's still a bit technical.