A chain of events occur when you move your finger across the rim of a glass and make sound. First thing you need to know is that material has a frequency at which it vibrates. This is called its resonant frequency. Your example of singing glasses is a good way to explain resonant frequency.

If you run your wet finger around the rim of the glass too fast, the glass will not make any sound. And if you run your wet finger around the rim of the glass too slowly, the glass will not make a sound. In order for the glass to make noise you will have to move your wet finger around the rim of the glass at the same frequency (or speed) as the resonant frequency of the glass (or material). Carefully, try this at home with glasses to see how fast you need to move your finger around the glass in order to make noise.

When you run your finger around the glass, this gives energy to the molecules in the glass making them vibrate. When individual molecules in the glass vibrate it makes the whole glass vibrate. This vibration then transfers to the sir molecules and that is what makes up the sound waves that you hear.

When you change the amount of water in the glass you are adjusting the mass of the system (the water and the glass). The more mass you have in the glass the less the glass molecules vibrate, the less the molecules vibrate the less energy that is in the system, the less energy causes the air molecules to have a lower frequency in which you can tell from the pitch of the sound.

What we call sound is a vibration in air or some other substance. The glass makes a sound because moving a finger along the edge makes the glass resonate, which causes the air inside the glass to vibrate, creating a sounds wave. Here is a web site I found with more info:

singing glass

It seems that adding water to the glass increases the vibrating mass so that the resonant frequency goes down and, thus, the pitch of the sounds wave is lowered. How do you think the pitch would compare if you used a bigger glass?