How can one prove that when matter changes physical states, its mass doesn't change?

You can perform an experiment to prove that mass does not change when a solid changes to a liquid (or vice versa). 

The easiest way to do this is to use a scale and an ice cube. You will also need a small dish to hold the ice cube. Use the following procedure:

1. Place the dish on the scale and zero the scale.

2. Put the ice cube in the dish and record the mass.

3. Wait for the ice cube to melt and record the mass. 

You should observe no difference in the masses you recorded. 

A similar procedure can be used to demonstrate that mass is not lost when a solid turns to a gas. You will need a piece of solid carbon dioxide (commonly referred to as 'dry ice') and a balloon. 

1. Place the balloon on the scale and zero the scale.

2. Place the dry ice inside the balloon, tie a knot to close the balloon, and record the mass.

3. Wait for the dry ice to sublimate (turn into a gas) and record the mass.

Again, you should see no change in the masses you recorded. 

These observations prove that mass does not change when state changes.