You've given us a one word diagnosis - "stiff" - but with only that it's hard to guess what your instructor means. If you're confused by the description, then whatever you're doing wrong is probably relatively subtle.
It's at least as likely to be attitudinal (i.e. your mental focus, as in trying to hit too hard, or "clenching" for too long, or rushing too fast through the early stages stages of the kick) as technical (e.g. not rotating the back foot to the right angle) or a physical limitation (flexibility, strength).
One thing to do is watch carefully what your instructor and others do, and see if there's any differences you can notice in what you yourself do. Does their technique seem less "stiff" to you in some way? If you can't work it out - ask the instructor what they mean.
As is often the case in martial arts, you can work backwards from the goal to the start of the technique. To do so, try a loose, light, relaxed kick with the front foot into a bag - just aim to get the timing right so it snaps satisfyingly into the bag without any conscious effort. Then gradually stand more square on, with the foot coming off the ground increasing from the side and back, so you have a little more turn into the kick, and use that turn to increase the power by snapping the hip around more strongly, dragging the knee and foot behind so it's some combination of centrifugal force and a whip like action when first the knee and then the foot snap around into the target. Focus on how much more angle the hip has to turn through, and position your feet and knees so they're snapping your hip strongly through that angle as your kicking foot leaves the ground. As you keep increasing the amount of turn you make into the kick, you'll end up with a back leg kick. If you do it right, I believe you'll find the back leg should be angle 45-60 degrees from forward, not 90 degrees as some arts/texts recommend, which prevents the back leg driving the hips powerfully through the kick and can easily result in a large, lumbering kicking action based on sustained muscular effort that will feel still and sluggish.
Another good exercise is kicking into the bag with your knee - but still have it moving sideways in the direction you want your foot to travel it during contact for mawashi geri. This removes the leg extension from the picture, giving you more focus on the hip and body rotation and the way the knee if accelerated snappily towards the target. Once that is all working well, adding the lower leg extension for mawashi geri is easy. The trick to a really strong, snappy mawashi geri is to feel like the the knee is left behind the hip as long as possible, then sling-shots past it as late and hard as possible. Vary your timing, stay loose and experiment.
Separately, you also could try practicing the kick and immediately afterwards moving loosely into other techniques (e.g. try a wide variety of kicks and hand techniques with both sides of your body) - developing the ability to flow smoothly, powerfully and fast between techniques will help you be aware of any stiffness, as stiffness generally slows you down and compromises your balance.