There are two parts to my answer - the philosophically floofy part that I will attempt to explain and the more serious 'how to improvise' part:

The floof
Improvisation is probably one of the coolest feelings that you can experience as a musician.There are analogous feelings in real life that can help explain - moments of uniqueness,creation,surprise like 2 people having the same thought at the same time or sense of achievement when you try something new and succeed or when you're in the "zone" while running,coding,gaming etc....

How-to
There are many...many approaches to improvisation and many of the answers have outlined a few good ones. The approach that I tend to take involves the following:
(1) you hear it in your mind as/before you play it - this is a trait ive found in many great improvisers, a lot of them can sing the notes as they play them. SCAT SOLO!!! As someone mentioned, what you hear over a chord progression changes over time.

(2) you reproduce what you hear in that moment - this requires that you build up your technical skill to the point where you can respond to your thoughts and play what u hear.

(3) Many improvisers are very theoretical with their approach, learning what scale/chord/arpeggio combinations. I personally have found that this approach has merits but its not for everyone. A basic level of theory is definitely required(triads, major scale, intervals), once you have this down, you can experiment with various sounds and modifications to these rules and many times, you come to the same point. At the end of the day, its the sound that you have to control. Your method is your own choice.

(4) With reference to point 3, I have found that with the 'play with your ear' approach, its clearly not all encompassing and you tend to miss a lot of possible sound combinations - my answer to this......transcribe!! When you hear moments you like, figure out what combination and sequence of sounds caused this moment. Once you figure them out, make them your own. You could copy them verbatim or you could use it as a building block to build your own. The important thing is to associate with the sound and build up your sonic vocabulary.

(5) mistakes are not necessarily a bad thing, ofcourse this is highly context dependant. There is bad playing...and then there is restrained experimentation. A mistake can be turned into a sonic gesture...a simple way that I use sometimes is to repeat and embellish a mistake, using the mistake as tension/buildup to the resolution.

(6) Many tend to focus on the melodic/harmonic aspects of improvisation and tend to forget aspects like rhythm,phrasing etc. These are equally important in improvisation - a whole minute of 16th notes tends to get boring, no matter what the melodic/harmonic context.

(7) dont be afraid to cross the line, walk the edges of your safe zone - its a thrill!!While you use the rules of music as your basic, breaking the rules is probably the most fun part of improvisation. Outside notes, rubato playing,silence are all improvisational devices.

Try some of these out and you'll find out what improvising feels like.