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Because the reflection is off of a point where the string is not allowed to move.

You can determine the phase of a reflected wave relative to the incoming wave from the freedom of motion of the point of reflection.

Since the point of reflection in the case of a string is not allowed to move, we can imagine that if it was a point on a longer string (and thus free to move) what would be the required phase of a wave coming from the distance moving in the direction of the reflected wave that does not move that point. Thus the reflected wave must always cancel the incoming wave at the point of reflection, the only phase that will fulfill this condition is the inversion (180 degree phase shift).

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