If we define an angle to be a measure of rotation, with the initial side of the angle corresponding to the direction in which we initially face and the terminal side corresponding to the direction in which we face after the rotation, we must be able to calculate it without reference to the size of a circle (since we are considering rotation around a single point).
We know, geometrically, that the circumference of any circle is $2\pi r$ giving us the constant of $2\pi$ for what corresponds to a 'full rotation' when we divide out the (potentially misleading) size (radius) of the circle. Once we have defined the unit of measurement for rotation (based on a 'full rotation'), we then define larger and smaller rotations as fractions of a 'full rotation', i.e. as the same fraction of $2\pi$. This correspondence factors through the computation of arclength/radius, similarly giving us a constant measurement (regardless of the length of the arc) for the rotation corresponding to the angle through which we rotated.
To answer your question, radians are just a particular way of measuring an angle by measuring a length which is a fixed distance away from the point of rotation in such a way that the fixed distance does not affect the measurement of the rotation.
Going further, it is clear that angles cannot be measured by the amount of "space" between the initial and terminal sides, since the length those sides could provide more or less "space". Also, rotation cannot be measured with a ruler, since we are standing on the same spot and rotating - there has been no distance travelled. Thus we must project our rotation outward from the point of rotation (radius), in order to have a distance to measure. But, we do not want that radius to affect our measurement of rotation, and we can already see that projecting a radius of 2 feet gives a different distance travelled (arc length) than a projected radius of 1 foot gives for the same amount of rotation. These differences in arc length are solely based on the radius of projection, and when that radius is divided out, the ratios all end up equal for the same amount of rotation, regardless of the radius we use to measure. Thereby, we have a consistent method of measuring rotation.