Why is an object moving in a circle at a constant speed said to be accelerating?

First of all, it is important to remember that an object's acceleration is the rate of change of its velocity, not its speed. Velocity is a vector quantity and so must have a direction as well as a magnitude. As the direction the object is travelling in is changing, its velocity is also changing and so it must have an acceleration even if the magnitude of the velocity (the speed) is constant. This acceleration is perpendicular to the velocity of the object at any given time and points towards the centre of the circle.