Green’s Dictionary of Slang

eat cheese v.1

[SE eat + SE cheesy; i.e. the quality of one’s smiles]

1. (orig. US black) to toady to, to ingratiate oneself with.

[US] in DARE.

2. (US) to inform on, to betray; thus cheese-eating n.

[US]J. Blake letter 15 April in Joint (1972) 19: To report these unsavory and totally lost creatures [...] is called ‘ratting’ or ‘cheese-eating’.
[US]B. Schulberg On the Waterfront (1964) 115: People I may know . . . you mean eat cheese for ya?
[US]H. Rhodes Chosen Few (1966) 125: It’s only a matter of time ’til somebody gits hip and eats cheese, baby.

3. (US black) to be in love.

[US]Current Sl. V:2.