A particle is under acceleration when it's speed changed over time.
When a particle is moving around a circle with constant speed.
The velocity is changed constantly. The direction of the velocity vector is changing with time. (Velocity is a vector.)

You can click Play to start the simulation and click pause to stop it.
More information will be displayed when the simulation is paused.
When the simulation is paused at time t: The RED arrow shows the velocity vector at that time.
Another vector in Magenta shows its velocity at previous time step t-dt.
(The starting point for the above two vector is different)

We also draw those two vectors again from the same starting point (which is the center of the circle).
You can find out the difference between two vectors (GREEN vector).

Click STEP button to make the time step forward and watch the differences.
I hope this applet can help you understand why acceleration is required for constant speed circular motion.

VIEW or DOWNLOAD ejs xml source
then click to translate strings for selected locale.

Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop.