I believe that MattDMo's hypothesis is incorrect. Only one group of ants, the Attini tribe, cultivates fungus as a food source. This group is exclusively a New World group, thought to have originated in the Amazon rainforest and spread out from there. I see from your profile that you are located in India, which is outside the range of the fungus-cultivating ants.
So if the ants aren't using your fingernails for fungus cultivation, what are they doing with them? Fingernails are primarily composed of keratin, a very tough protein which ants cannot digest. However, it is likely that the clippings also contain materials like dead skin, nail fungus, or other organic substances that the ants can eat, making them worthwhile for the ants to collect.