When Ponyboy says that Two-Bit is "grinning like a Chessy cat", he is comparing him to the character of the Cheshire Cat from the Lewis Carroll story Alice's Adventures in Wonderland. The Cheshire Cat is a feline with the magical ability to become invisible except for his toothy smile; oftentimes in the tale, the large grin is all that can be seen of him. Two-bit, who has just played a prank on Ponyboy and Johnny by impersonating a threatening Soc, is grinning from ear to ear, and his demeanor reminds Ponyboy of the famous cat from Wonderland.
Ponyboy's use of the allusion and while mispronouncing the term (Cheshire/Chessy) illustrates through marvelous juxtaposition the two opposing and seemingly contradictory sides of his nature. Ponyboy is a tough, unrefined member of the Greasers from the wrong side of the tracks, but at the same time he is highly intelligent, studious, and well-read.
A cheshire cat is a cat that has a huge grin and jokes from the story "Alice in Wonderland"--it is an allusion.