"To break the ice" means to, in a social
setting, be the first to talk about or do something to break down people's reservations
and get the conversation
(or some other activity) going.
Our present figurative sense of the phrase came from the maritime necessity of breaking up the ice upon rivers and channels for the navigation of ships in winter. The current meaning derives from a slightly newer metaphoric use of to break the ice, "to prepare a pathway for others," from the late sixteenth century. The earliest written instance of this intermediary usage is from A Briefe Treatise of Testaments and last Willes (1590) by Henry Swinburne:
"The author therefore in adventeuring to breake the yse to make the passage easie for his countrymen, failing sometimes of the fourd, and falling into the pit, may seeme worthie to be pitied."
The current usage of the term did not occur in writing until at least 150 years later.