In the land of Westeros seasons can last many years.
The phrase, "Winter is Coming", is a constant reminder that there will be a time when you will not be able to grow food, people will die of both frost and famine, and, above all, time is fleeting. It is the motto of the Stark Clan.
Another common phrase, at least in the books and in the first season, is "These are the acts of green summer men". Which, in my opinion, means the care-free nature that comes with a time of plenty has not yet needed to be shed. When you live in plenty you don't always think of the times of less. The Stark words seem to be "remember that there will be a time that we will need to band together for our own survival".
This significance is also increased with the threat of the White Walkers and the Others coming to attack. begin speculation These beings are basically the embodiment of death and the other spoken of by the Red Woman.