It means to be fed-up, bored, or angry, but not quite fed-up, bored, or angry enough to go on a rampage with a wet mop. It's more the sort of feeling that needs to be relieved by a long whingeing rumble of a moan.
It's how you feel when you have two essays to write; when it's raining again; when he leaves the apple core on the coaster for you to clear up for the third night running; when the fox keeps on pooing on the lawn; when yet again the supermarket has moved all the tinned tomatoes.
Yes, it takes a bit of time to get cheesed off, but all the same it seems that some people actually enjoy it - and some people (comedians, journalists, politicians) make a career out of it.
The trouble is that the approach of the habitually cheesed off is likely to make people run screaming.
So, on the whole, a slightly amused equanimity might be a better attitude to cultivate.
Thing Not To Be Today: cheesed off. This word doesn't seem to be anything to do with the food product but perhaps with the verb to cheese which means to stop (as in cheese it!) or, in prison slang, to grovel.
Mind you, the connection with those last two meanings isn't that obvious, is it.
Eric Partridge's guess is that this sort of cheese is the same word as cease, which comes from the Latin cēdere, to yield.