A mass is attached to a rope, and put into a circular motion. ... I am applying a force only in the radial direction, so how can the tangential velocity increase if there is no tangential force?
For short, I'll call the mass attached to the rope a "rock". So how does the rock gain angular velocity? If you truly are applying a purely radial force, and if you aren't shortening the length of the rope between your hand and the rock, the answer is that it can't. Radial forces do zero work. If you keep the length of the rope constant and you apply the force radially toward a fixed location, the angular acceleration of the rock about that fixed location is identically zero.
The rock does gain angular velocity as you swing it, and presumably you aren't changing the length of the rope between your hand and the rock. That means you aren't applying a pure radial force when the rock's angular velocity is increasing.
Think of how you get the mass at the end of the rope (I'll call it a "rock" from now on) moving. At the start, your hand makes a big circle about your body / above your head. You gradually tighten up that circle your hand makes as the rock starts moving about you.
At any point in time, the acceleration on the rock is toward your hand, but that is not purely toward the center of the average motion of the rock, and it is not normal to the rock's velocity vector with respect to your body. You are imparting a tangential acceleration on the rock, and you are doing so without having to continuously shorten the length of the rope.