From Tyler Cowen:

I can think of three reasons.

First, there is a placebo effect.  For the Westerner/outsider, eating with your fingers seems exotic.  For (many, not all) South Asians, eating with your fingers brings back memories of family and comfort foods.

Second, your fingers are highly versatile and they are often the best implements for consuming these foods and blending together spices, condiments, and foodstuffs themselves.  There is a reason why humans evolved fingers rather than forks.

Third, and how shall I put this?  A lot of South Indian food is vegetarian and eating with your fingers adds flavors of…meat.  The fleshy sort.

Eating a dosa with fork and knife is a very different experience, for Tamil food on the palm leaf all the more so.

From the Comments:

Being a South Indian myself I add my two cents

1. South Indian food (The Lunch/Dinner) is essentially rice with a liquid (Sambar + Variations, Rasam, Curd) (No breads) The spices in the sambar need to mix with the rice and the mixture is not fully solid, A spoon might work but the fork is a disaster as spices do not stay on the fork. Also an important ingredient is the vegetable which generally is dry or with a thick gravy. The mix makes the taste heavenly, Also coconut being an important ingredient mixes well when the fingers are used.

2. Idli Dosas and the likes are snacks which again is eaten with coconut chutney and sambar (which is a liquid) The taste of the Dosa intensifies when it is soaked in Sambar/ Chutney and the taste is intensified when the fingers are used as the fingers allow the dosa to take the shape and absorb the spices better

3.Indians prefer using the finger!!!! We are bought up that way

-Vijay

Via Marginal Revolution