I am wondering if the origin of this step in the recipe comes from recipes written in India, and have been copied over into western book without asking this questions.
I do know that many times in India, rice is stored in sacks with some kind of insecticide (usually boric acid) applied to it.
So there's always a need to wash the rice before it's cooked..
So maybe this step comes from that..
cooking.stackexchange.com/users/4283/notthetup
Sometimes rice is polished using talcum powder, which is one reason you would need to rinse it.
Another reason would be to get rid of extra starch.
The extra starch cause the rice to clump and stick, which isn't what you are looking for with basmati rice.
Normally you just rinse it until the water runs clear.
cooking.stackexchange.com/users/2053/John Roberts
Rinsing is a matter of personal taste.
The idea is that it removes excess starch from the rice, which if left in can result in a 'gloopy' consistency.
I'm not sure why anyone would spend 15 minutes doing it though
- I rinse my rice about six times (fill the saucepan, tip it out, repeat).
cooking.stackexchange.com/users/4194/ElendilTheTall
Basmati rice is supposed to flow freely though, if it clumps, you are using it against the norm and should probably stick with a stickier rice.
Usually Basmati is used Pilau style, or in a Biryiani, then the grains are slightly coated in some form of fat (oil, ghee, etc), so the starch you would definitely want rid of.
cooking.stackexchange.com/users/3432/Orbling
I have never washed any rice before cooking and every grain is perfect and separate.
Perhaps its the way I cook rice.
I put a little oil in the pan, sauté for a few minutes, then add the water and salt, mix it and cover until its ready.
Always perfect.
cooking.stackexchange.com/users/59221/Char
I used to rinse basmati rice prior to cooking by steaming in a closed pot.
Then I got fed up and couldn't be bothered.
Nothing changed - the rice remains fluffy.
cooking.stackexchange.com/users/37782/rdt2
Every Indian recipe I see seems to insist on rinsing basmati rice before cooking it. Why, I have no idea, because I never rinse it, yet I can discern no difference between my rice and rice cooked by other people that have, presumably, spent the extra 15 minutes rinsing.   If the stickiness is the difference, as suggested in  this thread , then I would suggest that  not  rinsing the rice would make eating rice-based thali a lot less traumatic because the grains would clump together more easily.
Why rinse basmati rice?
cooking.stackexchange.com
<rice><indian-cuisine>
cooking.stackexchange.com/questions/14738
cooking.stackexchange.com/users/4777/Doug
In India, the rice is stored in sacks with insecticide, talcum powder  or some kind of fat, which should be washed away before cooking. Rinsing also helps to get rid of extra starch, which would otherwise produce a 'gloopy' consistency. However, not washing rice in the West often does not have the same risks. Rice is sometimes treated with boric acid so needs rinsing. It also removes excess starch. 
In India, rice is treated with boric acid or polished with talcum powder so you need to rinse it. Rinsing removes the excess starch. It doesn't make any difference if you rinse it or not.