Beaver is grounded on a Saturday because he got into his father's desk when he knew he was not supposed to and caused some ink to spill. His parents then go on an outing, leaving Beaver in the clutches of Larry Mondello who tempts Beaver into going to the movies with him by enumerating the various attractions showing that day. What Beaver is not expecting is to win an expensive bike in a raffle that the movie theater is having that day, and now he has irrefutable evidence of his crime.
There was an easy solution to the problem - Larry could have claimed he won the bike when he went to the movies and left Beaver out of it entirely. Or, even more to the root of the problem, Larry could have not insisted that the reluctant Beaver claim his prize. Instead, Larry does take the bike home with him, but makes up a ridiculous story about a millionaire driving around giving free bikes to kids. Dean Wormer of Animal House had a few words to say about kids like Larry, but it will be a few more years before Larry gets to the drunk part.
There was a previous episode where Larry and Beaver skip school and end up on a local live TV broadcast with Wally and June watching at home. This seems to be a rework of that episode. I guess you can't have an original episode each and every week when you do 40 episodes of TV a year.