I believe the primary reason is that the pasta water is already hot.
When you need to thin your sauce on short notice, you add hot pasta water and it will not cool down your sauce.
Secondary benefits are: The pasta water has nice salinity, so you're not diluting the salinity level of the sauce.
This assumes you salted your pasta water.
You did, right?
There is some starch in the pasta water.
However, since the pasta water has already achieved a high temperature, any thickening benefit would already be achieved.
If your pasta water is anything like any I've ever seen, it's just about as runny as any other water.
So this benefit is probably negligible.
Perhaps it has more of an effect once the pasta water added to the sauce has reduced.
cooking.stackexchange.com/users/7137/Jeremy Ross
Adding water will thin a sauce, but the starch in the water does help it cling to the pasta, and adds some body to the sauce.
Another key step is to finish cooking the pasta IN the sauce (in a skillet, usually) before serving, allowing the starchy pasta to absorb the sauce more completely.
See also: http://www.seriouseats.com/2014/05/does-pasta-water-really-make-difference.html
cooking.stackexchange.com/users/7060/Jennifer S
In a restaurant you will cook a lot of pasta in the same pot of water over the course of the night.
This water will end up having quite a bit of starch.
Using a bit of this starch water will help the sauce adhere to the pasta.
At home the resulting water isn't going to contain as much starch, but it will still help.
You wont see quite the same effect as a professional kitchen.
I'm not sure which professional chef said it, but he said that it would be great if they bottled this starchy pasta water for home use.
cooking.stackexchange.com/users/7165/devin_s
Harold McGee did a great piece on the amount of water used to cook pasta that discusses this topic. http://www.nytimes.com/2009/02/25/dining/25curi.html
Basically, if you're going to do it at home, use less water, and you'll have a more concentrated, flavorful liquid, which you can use like stock in a sauce (not saying it is stock, but it has a lot of flavor).
It's obviously starchy and salty, but it will have a good amount of wheaty flavor.
I find the water from whole wheat pasta also has a lot more flavor than white pasta.
Probably the most famous dish that I'm aware of
that makes use of this technique is Cacio e Pepe, a Roman pasta dish where the sauce is made from olive oil, pasta water, pecorino romano and/or parmigiano reggiano, and of course a good amount of black pepper.
Sometimes butter is used as well.
It's a pretty cheap meal, but very comforting.
Sometimes I make a vegan version with olive oil, earth balance, fresh garlic and some nutritional yeast
(fiancee is allergic to dairy).
cooking.stackexchange.com/users/5776/paul
The starch in the cooking water acts as an emulsifier, so for a dish like spaghetti, aglio e olio it will make a more luscious sauce.
cooking.stackexchange.com/users/7079/Stefano
I thought it gives either the pasta or whatever you add it to flavor.
Once you cook the pasta in it, it has the starch and the salt that might enhance the dish more when you let it steam and absorb into the food.
cooking.stackexchange.com/users/7117/user7117
Many people build it into to recipes as they feel it changes the mouth feel of a sauce.
Some chefs refer to it as adding silkiness, or a creamy finish.
This is subjective of course, but i think it does add something to the body of a sauce.
On top of this
it's already seasoned and hot
so is good to "loosen" a sauce that has been simmering for a while (as other users have stated).
I certainly don't think it thickens a sauce (the starch content isn't that high), but it tastes thicker... somehow.
cooking.stackexchange.com/users/2391/vwiggins
Pasta water does thicken the sauce...
you don't pour in huge amounts, just add bits here or there to increase flavour and to thicken with starch.
The main benefit of this is that the sauce will bind with the pasta.
If you pour in a huge amount, it will drown your sauce.
Source: I learned to cook pasta sauces in a Roman kitchen.
cooking.stackexchange.com/users/17225/Stronzo
Adding the pasta water does thicken the sauce & help with acidity.
I make a pasta dish my family loves & once in a while it doesn't turn out
right.
I finally tracked down the reason for this.
Every time I forget to add the pasta water, it comes out wet.
With it, it's smooth, creamy & all the pasta has soaked up the sauce.
I cook my sauce down & then add a small amount of the water (about 1/3 cup)
then cook it a little longer to let it thicken back up.
It doesn't take but a minute or two to get it back where it was.
I also make my pasta with a little less water.
The past water just makes the sauce creamy and luscious.
Give it a try both ways.
I'll bet you'll see & taste the difference.
cooking.stackexchange.com/users/26304/pat king
I've noticed that many Italian chefs add to their pasta sauces some of the water they used to cook their pasta.  What is the purpose of this?
Why add pasta water to pasta sauce?
cooking.stackexchange.com
<sauce><pasta>
cooking.stackexchange.com/questions/16982
cooking.stackexchange.com/users/7123/David Latt
Pasta water contains starch and acts as an emulsifier, 'loosening', thickening and adding textture to a sauce as well as helping the sauce to adhere to the pasta. The pasta water will thin out a pasta sauce and help it to thicken. It will make the sauce creamier.
The pasta water is already hot and will thin out a pasta sauce. It will help the sauce to adhere to the pasta and make the sauce creamier. It thickens the sauce and reduces acidity.