The history of washing for bread:[1]

The obligation of washing hands prior to eating food is not rooted in the Torah, and is thus not of Biblical status, but rather is Rabbinical. The Rabbinical institution to wash hands developed gradually from generation to generation. Until the time of King Shlomo, there was no institution to wash hands prior to any eating of food, and thus no one washed hands prior to eating as an obligation. Shlomo Hamelech instituted the obligation to wash hands prior to eating Kodshim.[2] However, all other foods remained permitted to be eaten without washing until the times of Beis Shamaiy. One of the eighteen decrees that were instituted by Beis Shamaiy in the attic of Chizkiya was that Kohanim must wash hands prior to eating Teruma.[3] However, it was only later that the Sages instituted to wash hands prior to eating bread.[4] This institution took place after the destruction of the Temple[5], in the times of Rebbe Akiva.[6] 

The reason for the institution according to Kabala and Chassidus:[7]

The sages instituted that one wash hands prior to eating bread as a protection for their decree for Kohanim to wash hands prior to eating Teruma, in order not to differentiate between regular Jews and the Kohanim.[8] The following is the Kabbalistic reasons behind this decree: 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] See Derech Hachaim [Mittler Rebbe] p. 127

Other opinions: From some sources, it is implied that the decree to wash hands for bread took place prior to the destruction of the Temple, at the same time that the Sages made the decree for Teruma. [Implication of Admur 158:1; Levush 158:1] Some Poskim even write that Shlomo Hamelech instituted the Mitzvah to wash for bread. [Chayeh Adam 36:1]

[2] Shabbos 15a

[3] Shabbos ibid

[4] Derech Hachaim ibid

[5] Derech Hachaim ibid

[6] See Eiruvin 21b

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

[8] Admur 158:1; M”A 158:1; Taz 158:1; Tur end of 158; Chulin 106a