(condensed from Ein Ayah, Shabbat 1:38)

Gemara: Rav Chama bar Guria said in the name of Rav: A fast in response to a dream is effective like fire for flax. Rav Chisda said: It should be done on that day. Rav Yosef said: Even if it is Shabbat.

Ein Ayah: A person with a healthy body and soul, who didn’t dull his feelings, is prepared to sense anything that the Divine Wisdom enabled him to discern when he strays from the proper path even slightly. Dreams are a moral weather vane for a person, which pick up on a person’s inclinations towards physicality. Just as the body will react when a foreign object intrudes into a dangerous place (e.g., coughing when food goes down the wrong pipe), the soul reacts to the burgeoning spiritual deficiency. In the physical realm, if a person ignores the signs of problems, he is likely to have to deal later with a problem which has developed into a well-entrenched danger. While initially it could have been dealt with easily, later on it can be very difficult to correct the problem.
The beginning of every moral fall begins in the internal state of the spirit. It finds expression in thoughts and feelings, which lower him from his previous level and cause him to harbor a more animalistic and crude frame of mind. That which starts with problematic thoughts continues into damaging actions.
Hashem gave man the characteristic that he is shaken by bad dreams that are connected to a fall in spirituality. If he is diligent in trying to right his path and stop his fall while only certain inclinations and tendencies are involved, then he has used the dreams in the way that the Creator intended them. Since most of these issues have to do with an increase in urges and physicality and related negative attributes such as anger, arrogance and jealousy, when the dream sets off the appropriate alarm, the person should respond by countering these tendencies. One counters increased physicality by accepting a fast to increase his spirit, which leans toward goodness and sanctity, at the expense of his body. This can destroy the root that set off of the inner alarms that caused the frightening dream and return him to righteousness.
A fast is effective for a dream like fire for flax because it consumes it in a way that leaves no trace. However, the treatment must be carried out before the problem entrenches itself in the person. That is the reason that the fast is fully effective only if it is done on that day.
Shabbat is able to raise the level of sanctity both on a general and on an individual basis. The enjoyment of eating on Shabbat is important as long as one can use the physical nourishment to also enhance the spirit. However, an upsetting dream shows that there is a lack of internal balance, as the physicality is becoming overly and dangerously powerful. In such a case, the pleasant physical feeling of enjoying food on Shabbat is not going to bring a real gain to the person while the spiritual deterioration is taking away from his spiritual happiness. It is therefore worthwhile to fast even on Shabbat and strengthen the spiritual side in a manner that brings the person real joy.