The poem starts out with a jolly (and maybe slightly drunk) guy complaining to a poet named Terence about his poems. He makes fun of how serious and sad his poems are, and says they give him "the belly-ache." He'd much prefer, he tells Terence, to hear something he could "dance to."
In the next section, the poet Terence talks back. He tells this guy that if he wants to dance, he'd be better off drinking beer than reading poems. Terence is teasing the complainer, saying that he'd better stick to booze if it "hurts to think." He reminds him, though, that even if the world looks better when you're drunk, the feeling never lasts. He backs this up by telling his own story about getting drunk and then sobering up again. So, he suggests, if beer only helps for a while then poetry will be more useful in hard times (and, he reminds this guy, there will always be hard times).
To drive the point home, Terence finishes by telling the fable of King Mithridates, who gradually developed an immunity to poison. The idea is that swallowing a little bit of sadness in poetry, a little bit at a time, can make you stronger and more resistant to the pain of life. So poetry really is good for something. Take that, drunk dude!