In reality, the moon doesn't change shape at all. It is orbiting the Earth at the same time as it is also rotating, and the Earth itself is rotating and revolving around the Sun. This is why stars move in the sky and why the sun and moon rise and set in different locations each day.
The only thing that changes then is not the shape of the moon, but the appearance of the reflected side - the part that looks bright - to those of us on Earth. We simply see a different angle of the reflected side each minute, hour, day, week and month. This gives the appearance of a "new" moon or a "full" moon, when in fact the moon's shape is just the same as it was before.
Changing shape of the moon, also called phases of the moon is caused by changes in the part of the moon illuminated by the sun facing the moon.
We see moon because of the sunlight reflected by towards the earth. At any time the sun is able to illuminate only half part of the moon. The other half of the moon does not receive any sunlight because of its spherical shape. Depending on the relative positions of sun, moon, and the earth, different proportions of the semi-sphere-shaped illuminated half of the moon faces the earth, causing the shape of the moon visible from the earth to change.
On a full moon day, when the moon appears as a complete circle, the earth is positioned directly between the moon and the sun. Because of this the illuminated surface of the moon fully faces the earth, making the moon appear round. At other the earth is positioned away from the direct connecting sun and moon. Further the earth from this line, smaller is the illuminated part of the moon directed towards, and visible from earth. On a new moon day the earth is positioned farthest from this line joining sun and moon. AS a result the moon is seen as a thin crescent only.