stick to = adhere to
You lick a stamp and stick it on an envelope. The stamp sticks to the envelope.
If you throw mud or dirt at someone, usually some of it sticks to that person. Figuratively, politicians throw mud (lies, bad stories, scandal) at each other, and some of it sticks.
Teflon is a special, slippery material that manufacturers use to line or coat the surface of cooking pots and frying pans etc - so that when you cook, the food doesn't stick to the surface.
If nothing sticks to you, people can say all the bad things they want about you but it doesn't damage your reputation.