Newton's 1st Law of Motion
A body will continue in its state of rest or uniform motion unless there is an external resultant force acting on it.
Since the body actually turns/rotates/moves in a circle, there is a resultant force acting on it.
Newton's 2nd Law of Motion
The rate of change of momentum of a body is proportional to the resultant force acting on it.
It travels a uniform speed but it's direction is constantly changing. i.e. there is a change in velocity over time and hence a rate of change of momentum, which according to Newton's 2nd Law, implies that a resultant force exists.
Since only a change in direction results from this resultant force, the direction of the force would have to be perpendicular to the direction of motion.
Hence there is a resultant force acting on the object, directed towards the centre of the circular motion.
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