How does the awkwardness go away? From my experience, there are often two main solutions: guys and dogs.
By "guys," I mean the most common link between the two families. (This example can be extended to include two social groups that share some common members but do not often mix.) There is this pressure that guys (if they're the most "common" member of the two social circles) feel to help break the ice. Often times, girls play this role much more naturally. Everyone often looks to whoever it is organizing the event or joint outing for social cues as to what to do with the awkwardness.
And the best way to dispel it is through humor. That's where dogs come in. Having a dog around immediately (for better or worse) shifts the focus of attention away from the social interaction and onto the cuteness or hilarity of the animal. Everyone likes to pet it, ask questions about it, tell stories about its escapades, and that dispels awkwardness and also creates bonding (as the two social circles learn more about each other through their common experiences around dogs). Maybe cats and all pets help serve this function too.
But humor and cuteness need not be solely from dogs. Often funny hats and weird clothes can serve the same function.
I wonder if other guys (or pet owners) out there have felt the same way and what techniques others use to help their guests feel at home. When it's my job, I know that I feel responsible to do this as the guy, and my dog always comes to the rescue to help.