How would you explain to a non-Jew why we wash our hands 3 times after using the bathroom ?
I can't quite argue with @DoubleAA's comment about how to explain this to a Jew, let alone a Gentile. This article is a thorough discussion on the issue. You will see that - 1) It is debatable if one needs to wash at all. (Most would agree that one should, of course, for sanitary reasons, if not halachic ones.) 2 - If one does wash, does he need to use a cup?
I encourage you to read the full article, so that first, you can understand it, yourself. (Even after reading it, I have some trouble :) Only, then would I attempt to explain it to a Gentile. (There's little worse than trying to explain a halachic matter that you don't understand yourself to a Gentile. It would make you look foolish!) I'm excerpting what I think is the most relevant parts:
All the sources that mention a non-Beit Hamikdash-related washing after the bathroom are post-Talmudic. These sources include the Tashbetz, Kolbo and Mordechai, all of whom are cited by the Beit Yosef (OC 4). Other than cleanliness, there are two reasons discussed for the washing. Similar to washing before davening (OC 92:4), some view it as a preparation for reciting the berachah Asher Yatzar after using the bathroom (see Tur OC 165 and commentaries). Additionally, according to the Mishnah Berurah (4:40; 227:11) and others, merely entering a beit hakisei (an old-fashioned as opposed to a modern bathroom) imposes a ruach ra, which must be removed by washing. However, the mishnah regarding the Temple referred to above presumably refers to a Kohen who took care of his needs in a bathroom and yet the Talmud makes no mention of a ruach ra. Furthermore, many authorities argue that a modern, clean, multi-purpose bathroom does not convey a ruach ra even if an old-fashioned bathroom does.
The Aruch Hashulchan (OC 7:3) summarizes these sources as follows: If, while using the facilities, one did not touch anything dirty or any part of his body that is normally covered, there is no requirement to wash at all. Nonetheless, the custom is to wash, either because of cleanliness or because it is proper in order to say Asher Yatzar. He observes that some have the custom to wash thrice, but the Beit Yosef does not rule that way. The Aruch Hashulchan makes no mention of a keli or ruach ra.
According to those who hold that the washing is for cleanliness or in preparation for Asher Yatzar, there are few ritual requirements. Similar to the washing prior to davening, such a washing requires neither a keli nor three washings. According to the Mishnah Berurah (4:38), which holds that the washing is due to a ruach ra, the washing still has fewer rules than other ritual washings—such as for bread—and does not require a keli (Mishnah Berurah 165:2). The washing after using the bathroom is more akin to a washing for hygienic reasons, such as is required after removing one’s shoes or cutting one’s hair (Shulchan Aruch OC 4:18). Because the ruach ra of a bathroom is less potent than that which exists upon awakening (Sha’arei Teshuvah 4:12), only one washing is required for the former, while three are required for the latter (Magen Avraham 7:1).
Of course, when explaining to a Gentile, use the "KISS" method. Don't try reciting multiple shitot.
It is forbidden to teach a gentile this, for it is part of Torah and one may not teach a gentile Torah. The reasons for the mitzvos, minhagim etc. are all part of Torah, whether it is a mitzvah of the Torah, D'rabanan, (Rabbinical) or just a minhag yisrael. Minhag is also Torah.
So in answer to your question "How does one explain to a non-Jew...?" The answer is>> One doesn't!