Why do we say namaste with folded hands?

The phrase namas te is very old. It appears in the Rigveda, for instance at 8.75.10, where it is addressed to Agni, and it's a form of address you'd use for a god. Te is simply the singular “thee”, so grammatically it speaks to one being directly.

The gesture has its own history. Anjali, the two palms brought together, is described among the hand gestures in the Natyashastra, the old Sanskrit treatise on performance. In the Bhagavad Gita, at 11.14, Arjuna bows his head to Krishna with his hands joined (kṛtāñjaliḥ). In both places the hands are expressing reverence, not saying hello.

At some point this language of reverence became everyday courtesy, and we can't say precisely when. Today namaste or namaskar is the ordinary polite greeting across much of north India and Nepal, while plenty of Indians greet each other differently: vanakkam in Tamil, sat sri akal among many Sikhs, salaam or adaab among many Muslims. Namaste is widespread, but it's not the one Indian hello.

It has also become international, most visibly through yoga classes, and again in early 2020 when people everywhere were looking for a greeting that didn't involve shaking hands.

Folded hands work across a room or in a crowd, and you don't need to be introduced first. They let you honour someone without touching them. Some writers link this to a society that was careful about touch, including along lines of caste and gender. That's plausible but we can't prove it, and nobody knows that it's why the greeting took hold.