What do we mean when we say it today?
The bad part is over. We don't have to be unhappy anymore. That was rough. Now let's move on. That's basically how we use this phrase today. We use it to say that the winter is gone, and summer is here.
We Shmoopers now know that's not really what Richard is saying at all. He's saying the exact opposite: winter is now. His life still sucks even though his brother is on the throne. But we use it in a more positive way. We're really saying something like: "The sun is out at last! Let's play in it!"
It's also been used at the title for a John Steinbeck's novel, The Winter of Our Discontent, and a not so nice time in recent British history (that would be the 1970s) where a bunch of trade unions demanded more money.