"Whenever you fast, do not put on a gloomy face as the hypocrites do, for they neglect their appearance so that they will be noticed by men when they are fasting. Truly I say to you, they have their reward in full. "But you, when you fast, anoint your head and wash your face so that your fasting will not be noticed by men, but by your Father who is in secret; and your Father who sees what is done in secret will reward you. (Matthew 6:16-18 NASB)
Jesus is teaching about hypocrisy in Matthew 6:1-18 using the examples of giving to the poor, prayer, and now in this text fasting. It ought to be noticed that in the case of giving to the poor, there was an expectation that believers would do it, but a warning not to do it for their own glory. The same can be said for prayer and fasting as well. This statement 'whenever you fast...' makes the assumption that indeed you will fast. If the text read 'If you fast...' it would be altogether different. Now I am not an advocate for mysticism, and self deprecation, or piety for the sake of piety, at the same time the practices of fasting, prayer, alms giving, and other acts of devotion are critical to the Christian faith and they certainly have their place. If you give these things too high of a place you become like a monk and fall into the very sins Jesus is warning about in Matthew 6:1-18, but nonetheless pious action is not optional in the Christian's life. Fasting is not a requirement for salvation, but it is expected that believers will fast from time to time.
Jesus is addressing one of the greatest temptations in fasting, and that is to make it known that you are on a fast. There are a number of reasons we want to make it known, the first is simple pride. It seems that fasting is a discipline that is not often engaged in, so the moment you engage in one your flesh rises up against you and you find yourself believing that you have ascended to some level of spirituality that others haven't. Notice how Jesus gives the warning here, the hypocrites are not out saying 'Look at me I am fasting...' they are simply neglecting their appearance and putting on a gloomy face. If you have ever fasted for a significant time, the odds are that you have had these verse come to your mind and have focused on not telling people about your fast, but the real temptation is to act like you have a headache or wince now and then to get someone to ask you what is wrong, at that point you feel justified in exposing your fast. Jesus is saying that you must go out of your way to keep your fast hidden. Acts of piety and devotion are not tools for witness or even encouragement to the brethren, they are for showing Christ honor. When you have done these acts the rest of your life will witness to Him, because you have been placed in close proximity to Him through those acts, but the acts themselves should remain secret.
Now this super secrecy can become hypocritical as well. Jesus is warning against fasting publicly to bring yourself glory, at the same to being the super stealth faster can be just as selfish. If you are married, you need to tell your wife you are fasting! If you keep it secret eventually she is going to ask you why you won't eat, then you will tell her and in some twisted way you will find yourself glorifying yourself before her. Just tell her at the outset you are going on a fast. If you have a lunch meeting with someone let them know why you are not going to eat. Do not do it with pain on your face or make any sort of big deal about it, just tell them and move on. If you attempt to keep it secret eventually you will be approached with the question as to why you are not eating and then more attention will be drawn to you than had you just said something in the first place. When discerning whether or not to reveal a fast you must simply ask yourself if it would bring less attention to yourself if you reveal it up front, or if you can simply conceal it altogether. If you can conceal the fast without doing significant manipulation or giving half-truths then conceal it. That is the ideal situation. At the same time, realize that sometimes concealing brings more attention to yourself when you can no longer conceal it, than had you just revealed it in the first place.
Just a couple words on the benefit of fasting, the first and primary benefit of fasting is that it gives God honor by devoting your body in a tangible and physical way to Him. Fasting also gives you an acute awareness of Him, and of self, as the physical feeling of hunger as well as the purpose for your hunger constantly reminds you Him. Another benefit of fasting is that it shows you a sliver of who you really are apart from the grace of God. We realize that food is provision for us because of His grace. When withholding food from yourself as your mood gets worse and worse, and your head throbs a bit, and you find yourself feeling temperamental, realize that is a mere fraction of who you would be if God withheld grace from you. Fasting gives you a small window with which to see yourself as you really are.
Finally, realize that fasting is not a super spiritual thing that only elite Christians do, in fact I imagine more Christian than you realize are fasting, they simply do it obediently making sure you never know about it.