If you look close at the controller, the rubber at the top is quite porous and can retain quite a bit of moisture and your body's natural oils. When the controller is in normal use, you're always pushing on top of the thumb sticks so all that grime stays inside, but given a chance to sit, all this excess material starts to ooze out of the rubber.
Sounds gross, yes! Luckily oil and alcohol do not mix so wiping the sticks over with Isopropyl Alcohol removes all this grime.
I sprayed a good amount on one stick at a time (don't do both together as the alcohol will just evaporate before you manage to get to the second stick). Then, holding the stick to the side and using a cloth to wipe it down did the trick. I had to go over each of the sticks about 3 times (spary, wipe, repeat).
The difference was very noticeable after cleaning one of the sticks. The left one has been wiped down and you can see how shiny and oily the right stick is in comparison. The stickiness was significantly reduced after cleaning.
So there you go. Make sure you use something like isopropyl alcohol as you don't want to spill water on your controller and damage it. Since isopropyl alcohol won't damage electronics it's ok to have it spill on them.