Jazz saxophonist Charlie Parker, one of the eminent musical figures of the bebop era, is credited with having said, “You've got to learn your instrument. Then, you practice, practice, practice. And then, when you finally get up there on the bandstand, forget all that, and just wail.” This quote is one of my favorites with regard to the process of becoming an improvising musician, as it emphasizes the importance of strengthening all of one's musical fundamentals in order to improvise as naturally and musically as possible, and it does so in three distinct steps.
The first step in this process is learning, which is important to distinguish from practicing. The understanding of music in a conceptual and theoretical way, for instance, doesn't require practice, so much as study. The ability to identify where one's scales and chords/arpeggios are located on their instrument is very much the same way. So in this method, we would spend a significant amount of time finding these fundamental things in all of the ways they present themselves, including in all 12 keys. This sets one up to begin the next step of the process, which is where practice comes in.
In practicing, a musician takes something that he or she understands, and plays it repeatedly, to become more familiar with it. This familiarization allows the musician to slowly work up the speed at which they can play the thing being practiced, as well as reducing the stress required to play it. This is where someone develops a physical connection with their instrument, developing significant muscle memory on all their fundamentals that assists in the final step, which is when one can “forget all that, and just wail”.
It seems counter-intuitive to spend all this time studying and practicing just to forget about it when it comes time to perform, but this is arguably the most important step. It takes a lot of experience to hone this part of the craft, but over time, as one's fundamental connection with their instrument is improved, they can begin to play more from their intuition and by following their ear. As one studies and practices their fundamentals, they start to connect what they understand conceptually and what they have played with the sound that comes out. With thorough study and practice, an improviser can react appropriately to what another musician plays during a performance.
This is a general process by which musicians can develop the skills of improvising. The specifics as they apply to each instrument become more of an individualized problem to solve, and to solve these, it is best to find a very good teacher on one's instrument. Also, while this is mainly directed at those looking to learn how to improvise on a higher level, the concept of having very strong fundamentals on one's instrument is invaluable to their overall musicianship, so I am of the opinion that all musicians could benefit from this type of approach to practicing.