I haven't heard this expression is such a long time. I would interpret it literally. "Strike me" means "hit me". It is imperative mode. The speaker is saying "hit me in such a way as to make me silly". I would interpret "silly" as in the sense of "dazed". This would put the recipient of the blow into a confused state, rendering him unable to discern things clearly. So, when Mr. Owens says "strike me silly", he is saying that he is unable to believe what he sees, that he must be dreaming, or must have been hit over the head by someone who "struck him silly".

This is odd and/or dated language at best. Here's my guess. "Silly" can mean something like "simple-minded". If I just expand it slightly, I think he is saying, "May God strike me silly if, ...". Here, "may God..." is an expression similar to "I bet" or "I'm certain". Note that it isn't religious or indicative of any religious belief. It is just a turn of phrase in English, and people don't ordinarily say "May God strike me...", but I think that is the meaning here when expanded. He's saying he is so certain this is true that if it is not, let God turn him into a simpleton. A slightly different meaning of "strike me silly" is to literally hit someone until he has suffered mental trauma resulting in him being "silly"---not in his right mind. I don't think that is the intended meaning here. The only difference here really is who does the striking. If men do it, then they do it with fists or weapons. If God does it, then He Does as He Does. Eh?