Here let me spell it out for you.

You pick box 1. Guy eliminates a random box.

If box 1 has gold, you switch to a box with rocks, you lose.

If box 1 has rocks, you switch to 3rd box, you win.

If box 1 has rocks, you switch to 3rd box, you win again. (2 possible rock boxes to eliminate, so you have two outcomes)

Switching gives you 2/3 chance of winning, not switching gives you 1/3 chance of winning.

---------

Other way:

You random pick a box, 1/3 chance of winning

One box is eliminated.

You now have 1/2 chance of winning by switching.

Your chances 'improved' by 1/6, why would you not switch?