I came across a video by Maan Hamadeh, in this he has played Fur Elise in different styles. What is the theory behind this ?
There are many different ways to change the style of a particular piece. Fur Elise is a relatively easy one, because the chords can be heard very easily.
As you can see, it switches off from E7 and A all the way until you get to C. Then it goes to G, Am, and E (not shown). For what ever reason, the man in the video opted to go for F major instead of A minor in one part, but that is okay, because you still flowed correctly with the piece.
Understanding which chords go good together is nice, but it is not exactly need for what you call "playing in different styles". If you notice, in each of his styles, he plays the same chords with his left hand, and most of them stay true to the original piece.
The second part of your question asks how to change the rhythm. Well, that is kind of exactly what you do - change the rhythm. Note that in the original Fur Elise, the time is 3/8, but the picture I found with the chord symbols on it is in 3/4, so I'll be referring to that one.
For the first bar (after the pickup), the rhythm is extremely simple - just six eighth notes played successively. One thing you could do is "swing" the piece. You could play every two eighth notes like they are a dotted eighth and a sixteenth. Also, if you want to change it up, you could use octaves like the guy did in the video.
Some other things you could do:
- add ornaments (trills, mordents, etc...)
- add fast arpeggios in the correct chord
- change the chords (only if you know what you're doing)