"To cotton on" to something means to "catch on" or "understand" something. It is not commonly heard in today's speech except maybe in the southern US or by people from there. "To cotton to" something means to like something, but is almost always used in the negative. "He didn't cotton to his daughter's new boyfriend." It is more common, but still not frequently heard.

CORRECTION: the source for the dictionary definition was http://www.thefreedictionary.com/cotton+on

cotton on vb 1. to perceive the meaning (of) 2. to make use (of) Verb 1. cotton on - understand, usually after some initial difficultycotton on - understand, usually after some initial difficulty; "She didn't know what her classmates were plotting but finally caught on" ≡catch on, get it, get onto, get wise, twig, latch on, tumble ↔apprehend, comprehend, get the picture, grok, savvy, grasp, compass, dig - get the meaning of something; "Do you comprehend the meaning of this letter?" ----------------------------------------- I have never seen this phrase, but it seems to be a kind of "Baby Talk" or "Babyish Talk" of the kind that Dr. Seuss uses. In the Deep South of the United States, their are a great many such odd expressions. Here, it seems that COTTON ON is like a "slang" version of "CAUGHT ON" (as in UNDERSTOOD) Another word that appears from the Deep South dialects in the USA in the "Uncle Remus" stories, is the use of COTCHED for past tense, CAUGHT. Thus, if we understand a thing, we have COTCHED it or CAUGHT ON TO THE IDEA. So the common people are saying COT---ON to the Idea or COTTON the idea. Some of this is mere conjecture on my part, but this is how such terms are forged out of the experience of common folk. Another such term is where, in the Deep South, they refer to a person as very COONY. That is a reference to a clever animal, the RACCOON. So if they say a person is COONY, there are saying that he or she is very clever, in a sneaky and devious kind of way.

It's a cute, slangy way to state that you're "catching on" to something, meaning you're beginning to suspect and realise something that somebody's done or that you're slowly understanding a particular thing. Some examples: "Sooner or later, he's going to cotton on to your act." "I've finally cottoned on to Twitter." "Looks like she's cottoning on to our plan."

It means 'begin to understand' in the sense of eventually realising something about a situation. For example : 'We knew that Ted and his secretary Jenny were spending a lot of time together, but it took us ages to cotton on to the fact that they were actually having an affair.'