Improvising doesn't come easy to everyone, but it is a skill that can be learnt. I'm lucky in that I feel music emotionally and physically, so improvisation comes more naturally to me than others. However, I have to teach my students how to improvise. This is where I got a bit stuck.
They say "those who can't do teach" and this is actually true in some ways. People who have struggled and succeeded know the difficulties of learning and so are equipped with the various techniques that helped them. For me it is a matter of listening and feeling the music, knowing the general formations and phrasing, and anticipating them ... and being able to use my body to express it all. If I say that to my students they look at me with a blank stare. I've had extensive training in music so hearing the music is reflexive for me, and I have been improvising since I was little with my nursery rhymes playing on the record player, and my ballet music on cassettes and CDs. I can't expect my students to be able to have the same connection that I have. So I turned to the internet for help.
I found this article by Shira: How can I learn to improvise?
She talks about finding your "Now What?" moves and using them to buy time to think. She also explains the "Rule of 4s" in music (all music has this, although their are some exceptions of course) to help those who struggle with the music to count through it. Finally, she introduces the use of "Required" moves to be inserted intermittently once you are more comfortable.
I find this method to help my students get started with improvising. Then it's practice, practice practice.