Fasting at the time of Jesus had nothing to do with health, and everything to do with hastening the Kingdom of God by physically embodying one’s intense yearning for God to establish his will. Fasting also had to do with consecration for the High Holidays, like Yom Kippur. Overall, though, fasting is the physical expression of one’s intense yearnings. And this of course raises the issue of why many of us don’t fast — and the answer probably has less to do with our yearnings than with our disconnect of spirit and body. The fact is this: Westerners often separate the body from the spirit so radically than fasting, along with other physical acts, has been abandoned.

The attempt to recover such as “spiritual disciplines” in my estimation often (not always) misses the point. Fasting is not about the development of personal holiness so much as it is the inevitable manifestation in the body of what one yearns for in the spirit — it is a “I can’t go on, and I’ll show this by not eating, until I see God’s will come to fruition.” I could go on here, but I won’t. So, to sum up: fasting is the inevitable physical manifestation of heart-felt yearnings for those who have a better connection of body and spirit.
Yes, from what I can tell, Richard Foster and Dallas Willard see this connection but I think both are among the many who re-fashion this too much as a personal spiritual discipline — I see it as less of a discipline and more of an inevitable physical manifestation. The difference is this: as a discipline it is telic — that is, it is done in order to bring about something (holiness, greater discipline). As embodiment, it is not telic but consequence — that is, it is physical expression and brings the body into line with the whole person.
At any rate, Jesus is against fasting that is designed to bring attention to the one fasting. He counsels his followers to look like they do any other day so that no one will know what they are doing. Which of course raises the question of what you do when someone says, “Why don’t you want to go out for lunch?” The answer to that one is simple: “I’m fasting.” Why? The issue is not telling anyone; the issue is not advertizing that you are fasting.