What was the Brisker Rov's explanation for washing hands before Karpas by ורחץ?
Although we cannot achieve such purity today, we observe this law of "Netilat Yadaim" (washing) at the Seder in order or express our hope that we will soon be obligated to observe it, due to the arrival of Mashiach and the rebuilding of the Bet HaMikdash. This is an expression of the freedom that we feel on Pesach.
Another answer from the Netziv is everything we do by the Seder a memorial to the beis ha-mikdash. This is reflected in various minhagim of the Seder.Before eating Karpas, we wash netilas yadayim. All year long, however, the prevalent,custom is that we do not wash netilas yadiyim before eating foods dipped in liquids.Although the gemora clearly dictates that there is a rabbinic obligation to wash netilas yadayim,we rely on the opinion of Tosafos that this obligation was only in effect in the time of the beis hamikdash. Therfore the Netziv explains that we wash because of zecher le-mikdash
Although this is not answering your question, I felt obligated to write this information due to Yossi's one-sided answer. Washing before eating any food dipped and not dried off in one of the seven liquids that make food receptive to becoming ritually impure (wine, blood, oil, milk, dew, honey, and water) was required in the days of the temple for those who took on the stringencies of ritual purity. Shulchan Aruch (O.C. 158, 4) says one must still do so nowadays, but due to argument over whether it is required nowadays says to not make a bracha. Mishnah Berurah brings down that there are those authorities that still maintain that washing is not required but he strongly recommends doing so anyway without a bracha. Aruch Hashulchan says that most (in his corner of the world) are lenient and do not wash and he explains why there may be no reason to wash for dipped foods nowadays.The Netziv is of the same opinion. Sephardim are required to wash for dipped and still wet foods, but without a Bracha.
Fruits or vegetables dipped in water can acquire ritual impurity (Lev. 11:34). Washing before eating vegetables which have come into contact with water is a hold-over from Talmudic times. In that period many Rabbis attempted to eat all their foods in a state of ritual purity – trying to experience in their daily eating the sense of sacredness associated with the Temple. To emphasize that this is only a pious custom, and not even a rabbinic requirement, no blessing is recited.