When I improvise, I hear the entire next phrase in my head before I play it, so that when I actually play it, even if it is fast phrase, it can feel slow and relaxed: after all I have just heard it in my head much faster than I play it. This doesn't always occur, of course. Sometimes my fingers play things I have not heard first in my head. Those phrases are much worse.
The best phrases will feel very definite, very intentional, not tentative in the least.
Since I am not a very great improviser, I find it remarkable that I can do it at all. Extrapolating to someone far better than I am, I can see that they are hearing things in their head very fast and playing in a relaxed and intentional way when they actually come to play the phrase. It never seems like they are rushed to think of what to play.
The trick is not making things up on the spot, then, but making those improvised ideas sound inevitable in context. To achieve that I improvise hours and hours over a few different chords changes. That would seem to be the way to move forward.
We can all improvise in conversational situations. That's what conversation is, right? You might have a script for an interview or a sale pitch, but once you have to answer and unexpected question improvisation comes in. The sentence you are about to speak can also appear in your mind instantaneously before you speak it, so that the utterance itself comes out effortlessly. Or you can start to speak and flounder a bit, lose your place. Everyone can converse; some are better than others, but it is not a rare skill to have.