Velocity is a vector (it has size and direction). When an object moves in a circular motion with constant speed (speed = size of velocity), it is continually changing direction. Hence it's velocity is constantly changing.
Acceleration (also a vector) is defined as the rate of change of velocity.
Hence by changing direction, an object is changing velocity and thereby accelerating.
So in circular motion where we have constant speed but changing direction, you have acceleration. However, uniform velocity implies no change in direction, which is not possible if an object is moving in circular motion.
You can also bring this back to Newton's 1st and 2nd law. The first law states that a body will stay at rest or in uniform motion unless acted upon by an external force. The 2nd law tells us that the object will accelerate if a force acts on it (
i.e. you can have uniform velocity or you can have acceleration, but not both at the same time.