Using noodle cooking water to loosen sauce?

I always reserve a bit of pasta water to add it to the pan. The reason is simple: if you drain your pasta and add it to the sauce the pasta will suck up all the sauce and become a bit dry. Adding the pasta water ensures that your pasta will remain moist. Also yes, it helps thickening the sauce (this does not necessarily apply to tomato sauce).

Now, let's be clear: you DO NOT add your pasta water to the sauce when you are preparing the sauce. You only add it when you put your pasta in the sauce. It also helps if you don't drain the pasta, just take it out of the boiling water (using tongs, ofc) and put it in your sauce while still on the stove. A quick 30-50 secs stir and that's it.

Hope this helps.

PS: 1/2 cup seems like a lot to me. For two portions of pasta I usually add around 3 tablespoons.

In addition to the reasons covered in other answers, some pasta dishes with sauces including cheese actually require using some of the cooking water in order to turn out correctly.

In these cases the starch in the water coats the proteins in the cheese and prevents them from binding to the cheese's fat which would otherwise act as a sort of glue as it melts. This is the same reason why, when making Fondue, you typically coat the shredded cheese with a small amount of flour, cornstarch or arrowroot.

An example of a dish where using the pasta water is required for the recipe to turn out correctly is Cacio e Pepe (Spaghetti with Pecorino Romano and Black Pepper)

Based on a bit of goole centric research thekitchn.com: Quick Tip to Thicken Sauces with Pasta Water, Bon Appetit: How to Make Perfect Pasta tip #4, and a few others it looks like a common enough practice to add your starch water to sauce in order to give it a smoother creamier texture thus improving mouthfeel. You'd also be adding flavors from the starch water itself; presumable any salt you would have added to the cooking water. The recommendation seems to be primarily for oil based sauces, but may aid tomato sauces as well.

To me this seems odd, and I personally am not jumping to give it a go. I like my home made pasta sauces just fine thank-you-very-much.

They're recommending using the water in the sauce. It adds flavor, some of that being the salt you presumably added to the water. Also, see the answers to this question.