Why is an object that moves in a circular path accelerating when it has constant speed?

Think about the definition of acceleration - it is the rate of change of velocity. Your statement that the object's speed is constant is correct, can you think about why the object's velocity isn't constant? Velocity is a vector which means it has a direction as well as a magnitude, and the object's velocity is changing directiion (constantly) as it moves in a circle, therefore the rate of change of velocity = acceleration is non zero.