These answers explain the overall meaning, but they don't really tie that meaning in with the words. Saying someone is "crying in their beer" gives an imagine of someone seated at a table with a beer in front of them, and then they're crying and the tears are falling into the beer. It's actually an exaggerated insult. In English we often tell someone who is complaining that they are "crying". We exaggerate by implying they have become so emotional about their complaint that they are now crying. It makes the complaining sound extreme and serves to try to make the complainer stop complaining. It's actually a pretty mean thing to say. You say it when you don't want to hear any more complaining and you want the other person to shut up. You might say something like, "Quit your crying," or "Stop crying about it." So the idiom comes from the same idea. You are saying that someone is complaining so badly that they've been drawn to tears, and those tears are falling into their beer.

I was trying to think of an example. My wife didn't know I was answering an italki question. I live near Boston. The local football team is the New England Patriots. I remembered that they had an important game yesterday but didn't know who won. I thought I would use that in my answer, but didn't know who won. I asked my wife "how did the Patriots do yesterday?" My wife's answer--remember, she didn't know I was answering an italki question!--was: "Oh, they lost, so everyone is going to be crying in their beer today." She actually used the idiom! Suppose person A says "person B is crying in their beer about X." It means B is feeling sorry for himself. Person B is being self-pitying. Person A thinks that X is not very important. Person A thinks person B is being much too self-pitying. It's joking. It's teasing. It's mildly disrespectful. "I can't believe it. I just can't believe it. How could it happen? It was so close, so close!" "Oh, quit crying in your beer about it. The Broncos won. Tom Brady isn't Superman. Get over it."

It means to feel sorry for oneself.