Turf3 has got part of it, but there is also what you need to shut off. The creativity needs to come from you. It's you singing through your horn (as turf3 indicates). This means ignoring the normal "play the changes" mentality of just trying to impress with a whole bunch of memorized riffs and arpeggios. If you haven't got anything you can sing, then the next best thing is to have a big library in your head of those borrowed lines and do your "cut and paste" best.
So two things are necessary for REAL creativity: 1. have something musical to say, 2. being "one with your instrument" so that what you can hear in your head you can play with your horn.
The "mystical" part comes in after having achieved being one with your horn so that you don't consciously think about what you're doing. It is indeed like singing where you make sounds without thinking "now I'm going to sing/play an Ab followed by a D# ..." It's automatic with no translation through "theory". No REAL creative musician plays theory, they play music. Academics, who can't play, try to codify what creative people do with "theory". Theory isn't creativity. Singing through your horn is. Where it gets "mystical" is that you are not using your conscious mind. What's usually said is that the music "flows THROUGH you", or you can get really weird and say your channeling some dead guy or whatever. No weirdness or mysticism is necessary to achieve that state, but it does take a lot of practice to become one with your horn. Having something to say musically is another, and that's where you come to that word that nobody wants to say or hear: talent. The common attitude among students and teachers is that you learn all the prescribed stuff and you will somehow be a musician. Just isn't so. You can learn the cut and paste and use theory to remember how and were to do your cut and paste, but it's not being creative...and that's what you are asking about. Most learn to be musical mechanics, and if they are happy with that, then that's fine. To be a creative artist means that you have something unique to say and can communicate it through your instrument (or paint brushes) for others to enjoy.
What we usually hear from sax players is unfortunately the opposite, and this is partly the fault of the teaching establishment. It's a rare audience that wants to go see someone just trying to impress with a bunch of fast technique that says nothing. The player who can communicate and touch people's emotions will always have an audience. It a matter of having something to give and communicate. The standard of teaching and the way "theory" is taught is about copying players from 50 + years ago and doing the cut and past thing while playing tunes that nobody under 70 years old knows. They are taught to try and TAKE admiration for their technical skill rather than GIVE the audience something they can relate to. Is there any question why audiences are generally absent from strictly mainstream gigs?
Your question is a good one. Sorry about the long answer, but some of this stuff needs to be said. if sax players and teachers don't wise up they will find that their instrument will die with the type of music they are promoting. The clarinet died as a popular instrument with the Benny Goodman style. There is little evidence of mainstream jazz surviving as a part of popular culture. So will sax players and teachers continue to cast their own cement booties?