This is a fun one to observe! Water in solid, ice form actually takes up more space than water in liquid form. So, when we make ice in an ice cube tray, the water actually expands to a greater volume. Then, when the ice cube melts, it goes back to its original volume, which occupies less space.

Usually when things freeze, they occupy less volume and consequently their density goes up. Water is an exception to this rule:  when it freezes, it occupies more volume and its density goes down.  That's the reason why ice floats in water.  

A floating object replaces liquid equal to its own weight(Archimedes Principle).  For example, when a one oz ice cube is placed in a glass of liquid water, it replaces exactly one oz of liquid water. If we can take the ice cube out without disturbing the water, the "hole" can be filled up by exactly one oz of water. When this one oz ice cube melts, it turns into one oz of water filling up the hole filling the "hole." Therefore, no change in liquid level.  

Quick Experiment:  

Fill up a glass with water to its brim.  Put it in the middle of a big bowl.  Now take an ice cube, weight it, and gently put it in the glass.  Since the glass is full to its brim, the water will overflow into the bowl.  Collect the water and weight it.  The weight of the water should be the same as the ice cube.  This will test the Archimedes principle!

Cheers :)