Compensators are to reduce or eliminate the crew feeling the ship maneuvering. In previous versions of Traveller (Okay, I'm pretty sure it was in MT) you couldn't build a ship that had a higher G rating than what your compensators were rated for. So, at TL15, you could build a ship capable of accelerating up to 6G, because your G compensators could handle that much. At each reduced TL, you reduced the compensation by 1, so that at TL10 you had 1G of compensation, and I think they gave you 1G at TL9, just like you had J1 at TL9 and TL10.
TNE changed that a little, by allowing you to build to whatever G-rating you wanted, but you had to have a strong enough internal structure, and your crew might get turned to jelly if you accelerated more than your compensation. There was specific mention that you could exceed your compensators by 1G without ill effects, but each G higher than that, you needed to make rolls to see if you killed your crew or something like that.
And if you were in combat, this safety threshold was reduced by 1G, so you could maneuver up to your G compensation rating without ill effect, but above that, things got difficult. Not easy to walk when the ship is twisting and turning in all directions in an effort to make it hard for the enemy to figure out where you'll be when their laser beam reaches you.
As I recall, the progression of G compensation in TNE was the same as mentioned above.
There were some ways to get around this, to a limited degree. You could install special seats that would negate a G, or install G-tanks that would negate several G's; something like that.
I hope this is more helpful than my previous quip.