Why does Axii sign not work

You use your mouse to move the indicator inside the yellow bar back and forth. Move your mouse back and forth with the bar and make sure you keep the indicator within its constraints. If you stay inside the yellow bar, the bar moves to the right; going outside the yellow bar turns your indicator red and makes the bar move to the left. If the bar reaches the far right, you win, and if the bar reaches the far left, you lose.

As noted in the comments, your mouse controls movement of the indicator within the yellow area, not within the entire indicator, so you don't have to move the mouse much at all to win.

The strength, or difficulty, of your arm-wrestling opponent appears to influence two factors of arm-wrestling:

1. The initial size of the yellow bar. The smaller the initial size, the smaller the size of the bar will be when you push your opponent to the brink of defeat, and so the harder it will be for you to finish off your opponent.
2. The "resistance" of your mouse indicator. Much like how in a real arm-wrestle, your opponent will attempt to resist your arm movement, a stronger opponent in The Witcher 2 will actively try to force your indicator out of the bar at a greater speed and movements in the opposite direction will be met with tension, your mouse indicator visibly being pushed back by this apparent force. One of the biggest challenges of winning arm-wrestling is knowing how much you need to move your mouse to push your indicator back into the center without over-compensating. When you're about to win, the bar is very small, meaning the slightest overcompensation could move your indicator outside of the bar and give your opponent the chance for a comeback.

Because the bar becomes smaller the longer the arm-wrestle lasts, it's easy to return your indicator to the bar if it falls out at the start of the match but very difficult to return your indicator if it falls out when you were just about to win.

By default, you regain Vigor in combat albeit slowly. By bringing up the character stats page (default shortcut is 'c'), clicking "attributes", and then browsing to the "character abilities" tab, you can see many things, including your Vigor and Vitality rates (in combat / out of combat for both).

Specifically, Vigor regens at 1/3 of the rate in combat as it does out of combat (before modifiers)

Finally, you can't regen Vigor at all when you're under the effects of the Quen sign. (I have a feeling this might be your case).