It would be the same with proving that we live in a simulation, or that there is a god or gods, except as others have pointed out, we'd probably seek sandbox escapes.
Also, not every proof of being a simulation would necessarily be that. Any proof that our universe is a 4D projection of another with different dimensions might not include a proof of the direction of projection -- ours might be the true reality, and the other a projection of ours.
“we couldn’t break it or trancend it”
The same thing has been said about many man-made electronic systems, and we know how that’s gone so far ;) So I think for 95% of people, nothing would change, but for the folks on the forefront of experimental / particle physics, quite a bit would change because we’d be looking for a whole new class of exploits/vulnerabilities/ways to affect change in the simulation.
It would not have any sort of bearing on our lives, we couldn't break it or transcend it. Our feelings and our lives would feel just as real as before.