Trying to understand this jazz language before you have an understanding of basic chord progressions and tonal harmony is a bit like trying to read before you know the alphabet or trying to do math before you have the concept of individual numbers. Learn traditional harmony first so that you understand how jazz modifies the fundamentals to create a separate style. Make sure you understand jazz symbols, as these tell you what chords are going to come next in the jazz piece.
Much of jazz music involves incredibly fast melodic runs and development. In order for these sequences to sound soulful and natural, you have to be comfortable enough with the physical aspects of the scales and related sequences to play them virtually without thought. The only way to get to this point is through methodical practice--even if your mind is like lightning, your fingers might not move as fast as you think. Eventually, your muscle memory will take over, and you'll be able to sing through your instrument effectively. Learn all major scales first and then proceed to the minor scales and chromatic scales. As you work, gradually increase your speed, going only as fast as you can play the scales cleanly. Ideally, you will gain the ability to play 12 notes (a chromatic scale) per second.
Even improvised jazz has form. For example, jazz music typically is divided into eight-bar phrases, and both composed and improvised jazz melodies work around a given chord structure. You have to be able to continue through the form and chord sequences, recognizing where the melody needs to go. To train your ear and practice moving forward through the music according to the jazz form, listen to tracks by your favorite artists to see if you can identify why they improvised in a certain direction or what scales are used. Listen for the eight-bar phrases. When you feel comfortable with these, play a jazz chord progression and practice singing or playing whatever comes to mind (scat). Start with slow tempos first so you have time to think about what you'll do next. Then speed up the chord sequence. What you improvise as you listen doesn't need to be complex. It just needs to fit the chords with musical sense and feel natural. Worry about "filling in" your improvisations for more complexity later.