A few short lines set a small, funny scene: someone drinking ale at home, or somewhere equally relaxed, with only animals for company. The "monkey" is probably not a real animal. It sounds like the speaker's mischievous side, the restless part of the self that gets into trouble once the cup is in hand. So "my monkey and I" turns the speaker into a pair of cheerful troublemakers.
The last two lines are the joke. Once drunk, the speaker starts lecturing his hound in "the tone of an Aesop," like someone delivering a fable's moral. Aesop's stories often have animals teaching lessons, so the speaker flips it: the human preaches to the animal. He sounds wise and self-important, though the poem hints he is just tipsy.
The mood is light, wry and affectionate. The speaker laughs at himself rather than judging himself, and the title's "social drinker" becomes gently ironic, since his only company is a dog.