I think one of the easiest ways to improvise is to find some easy left hand accompaniment which allows you to play about anything with the right hand while still sounding at least ok. Here are a couple examples:
- With the left hand play the loop Eb, Gb, Ab, Bb, maybe about 1 note/second. I think the second lowest Eb on the piano sounds best.
- When you've played the loop a couple times, continue and also start playing any melody on the black keys with the right hand. Keep the left hand pulse steady.
- With the left hand play the loop D-A (as a chord), G-D. Again maybe 1 chord/second.
- This time play the melody on the white keys.
To make it sound more like "real music" try to at least have some breaks in the melody and try varying ideas, like how fast, how high, or how loud you play. Maybe take some simple rhythmic and/or melodic pattern and use that for a while. You can also every now and then play a white/black key which you "shouldn't" play just to see how it sounds. Usually it will sound good if you accentuate it a bit (so it doesn't sound like an accident) and then resolve it by playing a "legal" note after.
One way to improvisation is to learn a lot of recipes, and one way to create them is to just experiment. Theory of music in general, like knowing the basic physics of sound and the basics of how we make sense of music, will help your experiments go in a fruitful direction. Reading a book on composition or taking a composition class will probably teach a lot of useful knowledge in this. Studying and (actively) listening to lots of music will also help, and knowing the theory of the music you're studying will help you do it better. This is also a great resource for recipes. If something sounds good, take it into your vocabulary and then experiment to see what you can do with it. So in short theory is something that'll help you a lot. It's not required, though, as I believe for example many early jazz-musicians didn't know much or any theory and still managed to just play.
Theory is one side, and another is that you need to practice. You have to improvise a lot to learn to improvise. The two recipes I gave will get you started. Since these are quite easy, you'll be able to develop some important "universal skills". One is that you'll have time to listen to what you are doing and anticipate what you're going to do. You can also concentrate on creating music on the spot. By this I mean that you can concentrate on phrases, rhythm, dynamics, articulation and all that without also having to think about what the "correct" notes are. Don't underestimate this; even if you play everything "correctly", if your music is monotonic or chaotic or in some other way unmusical you just won't sound good.
If you think this is too easy or restricted you get 22 more recipes just by transposing these two. Try combining them. Some will sound natural after each other, some will sound wrong, and some will sound like they could work in some places but not in others. For example if you do the first recipe starting from Eb, then go to Ab, that'll sound pretty natural. Take A instead of Ab, though, and you have something which could be pretty effective when used properly. Theory will explain some of these but not necessarily all.
You get even more by changing the rhythm of the base loop. Or maybe play the notes twice as fast, repeating each one of them. Or play them twice as fast without repeating any. Maybe in the first loop add a short A before Bb, or G before Ab, or maybe something else. You can do surprisingly much with just these two basic schemes and some imagination! And by experimenting with these two you'll also develop skills in experimenting in general.
Maybe this still isn't what you are after but it won't hurt trying.