What was Big Willy Shakes going for?
As usual with Shakespeare, there are two meanings to this quote. The first is rather simple. Tranio is saying that if Petruchio breaks the ice with Katherine, or gets to know her, then he can woo her. This is pretty much how we hear this phrase being used today. If you "break the ice" in a room, then you're getting rid of the tension and everyone can be comfortable getting to know each other.
But the second meaning is much more specific. Tranio is implying that Kate is as cold as ice. She's mean, nasty, and despicable. No one wants to marry her or even be around her. So she needs to be broken (just like ice), so she can melt a bit. This goes back to the whole idea of taming that's in the title.
The Taming of the Shrew is not about the domestication of an actual "shrew" (the mammal Sorex araneus). Rather, the play's title refers to the way Petruchio breaks the will of the "shrewish" Kate. She's a stereotypically bossy, mouthy, and aggressive woman who is seen as trying to "wear the pants" in her relationships with men. Tranio is hinting at this type of breaking when he says "break the ice" to Petruchio. She may be a shrew, but she can be tamed, or broken—just like any other animal.
It's worth noting that originally, "break the ice" was a term used to describe ships. If someone broke the ice of a ship, he or she took it on its first voyage. Of course that use was really old school by the time ol' Shakey penned this play. This is the first time "break the ice" is used to describe a social gesture though. There can always be multiple meanings in Shakespeare, but it would be a bit of a stretch to think this was about a ship.