What was Big Willy Shakes going for?
Shakespeare is going for as many identity-confusing jokes as he can here. He's got two sets of twins to add to the mayhem in the play. Turns out the lost set of twins have been in Ephesus the whole time. Remember the boys also share names: Egeon's twin sons are both named Antipholus, and the twin servants are named Dromio.
As if this weren't confusing enough, S. Dromio is inside the house while E. Dromio is outside with E. Antipholus. When E. Antipholus hears someone talking from inside the house, he wants to know who it is and what is going on. He's being locked out of his own house!
As you can probably tell, there's a whole bunch of confusion about who's who, but that's exactly that makes this scene funny. The audience knows there are two sets of twins, but the characters don't think their long-lost mirror images are roaming around the same town (or house). Who's up for a little dramatic irony?
S. Dromio is simply saying he doesn't want E. Antipholus (or whoever he is) to get sick. If he stands around not moving long enough, he'll catch a nasty cold. Ick. Shakespeare introduced this idea of "catching a cold" to refer to someone falling ill.
Once Shakespeare came up with this phrase, he used it in everything from Troilus and Cressida to King Lear to Cymbeline.