Mantle is an API or Application Programming Interface. It will allow for far more optimization and use of actual processing power available to the GPU and CPU.
Most modern game engines using DX use about 50% pf the total CPU but do so in a way thats inefficient because of driver overhead and driver's doing a lot of the work rather than the programmer, making it harder for optimization and performance drop offs at rather low batch calls. Mantle comes in and allows for basically 100% control to the programmer allowing him/her to create as many batch calls as the engine can handle. Mantle will allow the programmer to do more with less, meaning he can do more with less processing power. This would allow for better frame rates, better visuals, and the biggest reason why its a game changer, Better games! Better games will come out with it because of the less restrictive overheads and etc. allowing for various improvements to games and such. For example a strategy game with 20k units on the screen each doing individual tasks or a space game with 50k spacecraft in an all out war.
I could keep going but too tired too
Also as DiceHunter said. AMD hardware benefits the most but Intel and Nvidia also get a boost, just not to the extent AMD chips will.
And regarding the 290x vs 780ti: The Sapphire 290 Tri-X when overclocked can get on the heels of the 780ti. No reviews are out for the 290x Tri-X card but if it acts the same as the 290 it will easily be the new king of the hill, stock or overclocked.
Edit:
As far as i know, Mantle will
run on Nvidia cards but the performance increases won't be much.
Intel CPUs do get the benefit though along with AMD CPUs.