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HalfPint is a trusted home cook.
I think it is all about flavor and maybe a little more color.
Flavor is a huge part of it. Because it is such a concentrated product, it gives you that deep, rich, "long-cooked" flavor without much work. It adds a savoriness and "umami" to a dish (mostly stews, sauces/ragu, etc.) that you can't get from many other ingredients. It doesn't necessarily always add a "tomato" flavor. It can make a quick-cooked tomato sauce taste like it took hours to make.
Another thing it does is thicken sauces and stews/soups. Tomatoes have a lot of natural pectin; a squeeze of tomato paste is the equivalent to about 8 or 9 tomatoes.
Great ingredient to keep in the pantry.
Flavor is the main reason for adding tomato paste to veal bones when making a veal stock. Its not so much the raw tomato paste flavor that you want; however, it is the carmelization of the tomato paste that adds the depth, flavor and color to a good stock or sauce.
pierino is a trusted source on General Cooking and Tough Love.
Actually I believe the real reason is to add a depth of color to your stock if you are moving on to brown sauce or demi. I personally don't get the umami or upapi and I've prepared this at least 100 times. You "paint" the bones with tomato paste while they roast with your aromatic vegetables and you get a pleasing brown color.