It might sound stupid but seriously I couldn't find why?
We have a mass with a constant speed, which is acted on by a unit force which is always at right angles to its direction of motion.
Why do the mass travels a circular path!?
As Newton did in his Principia Mathematica with an example similar to yours, imagine what happens over a short time $dt$ to the velocity of the mass. Using $F = m\,dv/dt$ gives
$dv = F\,dt/m$
The force is at right angles to $v$ and so $dv$ is at right angles to, rather than along $v$. Do the same for the next $dt$ and so on, adding up all the contributions of $dv$ for one complete revolution and you will get $\sum dv = 0$. The additional wobble to the circular motion becomes zero as $dv \rightarrow 0$
Assume the motion is confined to a plane. If the force is always at a right angle to the motion, the speed is necessarily constant. The circular shape follows due to symmetry.
To find the change in speed of an object, we can project the force vector acting on it onto the velocity vector. For example, the force below will create a large change in speed, because it is pushing mostly the same way an object is going.
This next force is just as big, but it will only create a small change in speed because it is pushing only partially in the direction of motion.
This final force will create no change in speed because it is perpendicular to the motion. Its only effect is to change the direction of motion.
Because the problem specifies that the force is always perpendicular to the motion, the object's speed is constant.
Once we know the speed is constant, it follows from symmetry that the motion is a circle. If the motion were, say, a parabola, then different parts of the trajectory would "feel" different to the particle. For example, the very bottom of a parabola is highly curved, but if you go up the sides a way, it is not.
The green part of the parabola, at the bottom, has a very short radius of curvature. The red part has a very long one. A ball traveling at constant speed feeling a constant force should not have such different types of motion at different times. Instead, since the force on it is always the same relative to itself, it should always be doing the same thing. There should not be any part of the curve that is different than any other part.
Other shapes, like an ellipse or a spiral, are also different in some parts than in others. Only the circle has the same curvature everywhere, and so the motion must be a circle.