A pastor friend once pointed out to me that Jesus says "whenever you fast," meaning that Jesus is assuming that you fast as a normal part of your Christian practice. Like most modern evangelicals it doesn't occur to me to make fasting a regular part of my routine. Worship, fellowship, sacraments, Bible reading, giving, serving, praying . . . and fasting. But maybe it should. I know that fasting, when combined with prayer, is a way of humbling yourself before God when you are particularly intense about making a request. Recall how Queen Esther asked the Jewish people to fast for three days as she prepared to risk her life to come before the king. I've also noticed that when I'm really, miserably sick, I seem to pray more effectively. There does seem to be a close connection between physical weakness and spiritual strength. I'd have to assume that the bodily weakness that comes with fasting strengthens your prayers in a similar way.
However, the point Jesus is making is not that you should fast--because he assumes you already are--but how you go about it. He says that when you go through your pious sufferings, you should not be making a big display of it for everybody to see. Don't go around looking all haggard but conceal your discomfort from others so that only your heavenly Father knows. You might even say he's asking us to practice a kind of "reverse hypocrisy." If hypocrisy is putting on a display of piety when in secret you're far from it, then this reverse hypocrisy is about having the appearance of normalcy when in secret you are making a sacrifice to the Lord that is somewhat costly.
I don't think he's talking about keeping your sufferings to yourself in general and never allowing others to bear your burdens. (We are commanded to bear one another's burdens after all.) Instead it sounds to me like he's talking about certain acts that you ought to be doing for God's eyes alone. Giving alms and praying fall into that category--which Jesus talked about earlier. And now fasting.
This is the third time in the chapter that Jesus refers to God as "your Father who sees in secret" and promises that "your Father who sees in secret will repay you." As I've mentioned earlier, sometimes I wonder if we've gotten way too caught up in thinking Christianity is all about "being an example" and "being a witness" and generally putting everything out there on display for others to (supposedly) benefit from. If we're not careful, we'll get all our reward here on earth through the praise of others, and there will be nothing secret left for the Father to reward. Besides, putting everything on display takes all the fun out of it. Secrecy gives you a chance to cultivate your relationship with God, to have something that belongs to you and him alone, making your relationship with him seem all the more real.