I guess it's because bossanova is basically light jazz, and it's jazz I can't wrap my head around
Sure you can. I used to think the same thing until I started practicing with some jazz musicians. To be fair, jazz theory can go really deep, but to get started you just have to get used to the idea that you have a chord progression that you're improvising over, and you can find a scale that corresponds to each chord. The chords in simple jazz standards tend to be mostly or all 7th chords.
Let's say the first 3 chords in the progression are C7, A7, and D7. You might pick the following scales to correspond with those chords:
C Blues Scale - C Eb F Gb G Bb C
A Mixolydian - A B C# D E F# G A
D Mixolydian - D E F# G A B C D
You can do pretty much whatever you want within those notes during the corresponding chords without sounding horrible. Beyond that, there is a lot of jazz theory to explore, but that will get you started. The Real Easy Book has several jazz standards to try out, and it is presented in a way that you can open up to one of them, see the chord progression and melody on the left page, and on the right page it suggests scales that you can use to improvise.