The reason for the institution according to Kabala and Chassidus:[1]

The purpose of washing hands prior to eating bread is to distance the unclean forces [Kelipos] from receiving nurture from the food one eats. All food comes from the Chesed of Hashem, and naturally allows everyone to receive from it, even forces of evil. This is even more applicable to bread, as bread is rooted in the tree of good and evil, and thus has ability to distribute more to evil than other foods. Accordingly, it is possible that when one is involved in eating food, which comes from Chesed of Atzilus, he will in the process feed the forces of evil, and be overcome with foreign lusts and desires. This is why we see that a person is naturally more prone to lusts for sinful behavior after eating a satiating meal. For this reason, the Sages established to wash hands prior to eating bread, which is the main satiating food of man, in order to prevent the forces of evil from nurturing from the Chesed of the food. This is accomplished through washing the hands, which contain the nails, which represents the level of the external forces of Chesed. Water, which is colorless, represents the level of Chochma, which prevents the Kelipos from nurturing. Now, why did the Torah itself not command one to wash hands, and why did the Sages establish the decree only after the destruction of the Temple? The reason is because the decree was made commensurate to the level of evil that existed at that time. Hence, prior to the destruction when revelation of G-dliness was readily apparent, there was no need to wash the hands to prevent the evil forces from nurturing from it. However, after the Divine revelation ceased after the destruction, the forces of evil increased in number and quality, thus requiring the institution of washing the hands prior to eating bread, as explained.

_____________________________________________________

[1] Derech Hachaim [Mittler Rebbe] p. 127