A direct allusion to the 1950's American sitcom, I LOVE LUCY. In the show, the American protagonist (Lucille Ball) has a husband (Desi Arnaz) who is Cuban-born, and who occasionally makes some unintentionally humorous gaffes with English. One of these is his curious pronunciation of the word "explaining", which he always pronounces "'splainin'".
A frequent theme on the show was his discovering an entangled deception of his wife's, whereupon he'd intone sternly to her: "Lucy, I think you've got some 'splainin' to do."
Today people use the phrase both as a fond allusion to a cherished TV show, but as a way to imply that another person is at fault with something and needs to come clean.
Example 1. When the rock group successfully showed that their record company had not promoted their poorly-selling album sufficiently, the suits at the company had some serious 'spainin' to do.
Example 2. My Rolex, my computer and my car are all gone... and you've got a bad heroin problem? I'd say you have some 'splainin' to do!