As classical musicians, we are trained to execute what's written in the music faithfully and stylistically. Going rogue is usually discouraged, which can make improvisation feel wrong, foreign, or like something exclusively belonging to the jazz cats.
But this couldn't be more wrong! All the ornamentation you hear in Baroque pieces started out as off-the-cuff additions to the melody, and the gorgeous, bombastic cadenzas found in concertos were originally a time when the soloist could show off their improvisation skills.
At the end of the day, improv is just you moving around in a key signature. That's it.
So how do you start? I like to take a simple scale, like C major, and put each note of the scale on an index card. Mix up the cards and lay them out on your music stand in whatever order they come out of your hands. Play through those notes, first evenly, then adding different rhythms as you get comfortable. Keep mixing up the cards and the rhythms you use, and voila! You're improvising.
As you get more familiar with this, choose one note that doesn't belong in the key signature (for example, B-flat when you're playing in C major) and add that note into the group of notes you're using. Keep adding "wrong" notes as you get comfortable.
I like to play a simple bass note on my flute for my students as they do this (for example, a low C if they're playing around with a C major scale) so that they can hear how each note they play relates to the key they're in.
So let's reclaim improv, classical musicians. It's just as much ours as it is anyone else's, and it's an incredibly liberating practice. Go forth and make things up!