Originally this phrase
literally meant to break ice, like ice breaker ships that would make
a path for other ships to follow.
During the late 17th century, the phrase took on the current meaning
of forging a path through an awkward silence or awkward social
situation.
About this time, ice-breaker also became common. It is a
mechanism, such as a party game to break up socially awkward
situations and to get people talking.