Recall that every vector has both magnitude and direction. In the case of the acceleration vector, the magnitude is how fast the velocity is changing and the direction is which way. For uniform circular motion, the acceleration is directed toward the center of the circle. The direction of the acceleration is continually changing because which way "toward the center of the circle" is, continually changes.
For example, consider a car moving counterclockwise (as viewed from above) on a circular track. Imagine that you are looking down on it from above, with east to your right. The center of the circle is always to the driver's left. When the car is at the eastern-most point of the circle, the center of the circle is to the west of the car, so the acceleration is directed westward. When the car is at the northeastern-most point of the circle, the center of the circle is to the southwest of the car, so the acceleration is directed southwestward. When the car is at the northern-most point of the circle, the center of the circle is to the south of the car, so the acceleration is directed southward. By induction, the direction toward the center of the circle for any given point on the circle (and hence the direction of the acceleration for any given point on the circle) is different than it is for any other point on the circle. Thus, the direction of the acceleration of an object in uniform circular motion is continually changing. That's why we have to be careful to say that, for an object in uniform circular motion, the magnitude of the acceleration is constant rather than making the incorrect statement that the acceleration is constant.