How does Newton's first law explain circular motion?

A Partial Explanation

First, consider the object moving in a straight line with no forces acting upon it. According to Newton's First law, that object will always move in the same direction, at the same speed, forever. This is inertia.

Now, for that object to move in a circular path, a force must act upon the object. If the force is in opposite direction of the object, the object will eventually slow down, stop, and begin reversing. The degree of slowing depends on duration and amount of force. Similarly, forces in the same direction of the object would increase velocity.

Since the question asks about "uniform" circular motion, we'll assume uniform in both speed and geometry; and a uniform speed shows that no force is acting in the same direction, or the opposite direction of the object.

So far, we know that there an object moving uniformly, and thus there's a force being acted upon it. Yet, we also know that force is neither "forwards or backwards." One more thing is the question asks for circular motion, not globular, so that makes this a two dimensional problem.

So if there are only two dimensions to choose from, and we've eliminated the one the object is moving on, then there can only be a force acting perpendicular to the path of the object. If that path is circular, then that force will always be towards the center of the circle.

Basically, think of the circle not as a long continuous path with constant force (even though it is) but as a whole series of little straight paths with a force knocking it to the side all the time, toward the center of rotation.