A fink is a tattletale, a stool pigeon, a strikebreaker or scab or union spy; in short, an unattractive person who weasels out of a bad situation by cooperating with authorities. "Finking out" is backing out of an obligation.
The word is said to come from the name of Albert Fink, who worked for the Louisville and Nashville Railroads from around 1860 to the late 1880's. Fink worked his way up from engineer to president of the company. His spies -- the original "finks"-- infiltrated the labor unions to report on their meetings and plans for strikes.
A ratfink is just like a fink, but worse.