Why did this aircraft re-route?
Its hard to tell from the picture, but assuming you are talking about the yellow airplane and the "wiggle" in the course, that was probably just an S-turn for spacing.  Sometimes the controllers don't get the spacing exactly right, or one airplane slows down early and messes things up.  In those cases, in visual conditions,  the controller can ask the following airplane to make an S-turn.  This causes the plane to take a bit more time to cover the same distance as flying straight and spaces the planes out.

This may be used instead of a speed assignment for a number of reasons


Planes on approach may not be able to slow down further as configured
There may already be a speed assignment but it wasn't enough
Slowing down takes time, and S turn may be more efficient.


An ATC instruction may assign a heading and follow up with a new heading and approach clearance or other instructions to join final, or they may simply ask for an "S turn to the right for spacing".  The vectors would be more likely in instrument conditions and the S-turn more likely in visual conditions.