It would be my pleasure. This practice is known as “mayim achronim'' (lit. “water afterwards”). The Shulchan Aruch (Code of Jewish Law) states that washing one's hands before birkat hamazon is an obligation. Rabbi Chaim Pinchas Scheinberg, zatzal, once told me that women are included to the same degree as men in this obligation.
There are two reasons which are offered:
Just as a soiled kohen is invalid to perform the Temple Service in the Beit Hamikdash, so too soiled hands make a person unfit to say a blessing.
To clean off any "melach sdomit" — "salt of Sodom" — that might be on the hands. Melach sdomit was a strong salt harmful to the eyes.
Why have you seen people wash before the meal but not afterwards? Perhaps it is since some people actually have the custom to not wash mayim achronim. Why not? Since melach sdomit is virtually non-existent today, and the concept of "cleanliness" is a relative matter and most people don't consider their hands "dirty" after a meal, therefore washing them would not be necessary.
I once heard a beautiful explanation of the symbolism of mayim achronim. This water washes off the "Salt of Sodom." The people of Sodom were infamous for their stingy cold-heartedness, especially regarding hospitality towards strangers. For example, the people of Sodom surrounded Lot's house and ordered him to send out the wayfarers he was hosting. After a meal, having eaten our fill, we might not empathize with a poor stranger knocking on our door asking for a little food. This quality of cold-heartedness is the antithesis of Judaism, and therefore we "wash it off" — saying: "We want no part of it!"
Sources:
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Chullin 105a,b & Tosafot, Berachot 53b and Tosafot
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Shulchan Aruch Orach Chaim 181:1,10; M. B. 22
