In days past, a dwarf told his king that he, too, wanted to be a knight. Too small, said the king. But the determined dwarf went about the kingdom, catching bandits and rescuing maidens, until the news got back to the king. "All right," said the king. "I dub thee knight."
Special miniature armor was hammered out for him. A galley knife was honed into a sword. But no strider little enough was found. So the king substituted a large shaggy dog. And the dwarf went forth again to do good.
A terrible rainstorm came up. The dwarf rode to a nearby inn. But the innkeeper said there was no more room. The dwarf pointed out how little space he'd take up. The innkeeper looked him over, and his mount, too, both soaked to the skin, and finally said, "Come on in. We'll find a spot for you. I couldn't send a knight out on a dog like this."
The punchline is the last sentence.
It's a play on the phrase, "I wouldn't send a dog out on a night like this", something people say when it's so nasty and stormy outside that it's unfit for any living creature to be outside if it can be avoided.
Sort of similar to "I wouldn't wish that on my worst enemy" implying something so awful that nobody should have to endure it, no matter how much you don't like 'em.