Why do people do the Doran's Blade stacking?

The only role that can viably buy an early Vampiric Scepter is a jungler. This is because lifesteal procs off auto attacks, and in early game you should not be auto attacking creeps because it will push your lane too much.

One of the main ideas behind stacking Doran's is that it provides a cheap advantage for early game harrasment. If two champions with similar base stats are harassing each other, the champ with 2 Doran's will have greater sustain in lane and most likely will win the harrasments. Having more health allows you to win the skirmishes against your opponent, survive jungler ganks, last hit easier, and stay in the lane after a failed jungle gank or harassment.

This does put you behind for getting your first big purchase item, however given that a B. F. Sword (the most common big purchase for AD champs) is 1550g, spending an early 475g on a second Doran's, and consequently getting greater CS (note that after ~25 CS the dorans has paid for itself) and champ kills, can actually allow you to get that B. F. Sword sooner.

The thing is Doran's blade provide health, damage and lifesteal benefits, all at the same time. If you open up with Vampiric Scepter because you don't have the bonus +200 health, you will be squishy, and you will be forced to return back if you are playing against a good harasser. If you open up with Ruby Crystal, you will be missing the lifesteal and replenishing lost health will be harder while staying on the lane. Doran's blade provides good balance between those perks.

And the main idea behind stacking Doran's items is to make difference in short term. If you manage to get a couple of kills, thanks to Doran's items, you will have enough money to buy something else. Stacking Doran's items are more suited to aggressive playing styles, short term high return investment. If you are playing for long term, it is better to be careful on the lane and buy something that can be combined later to get higher level items.

People buy them because they give you a big advantage in early-to-mid game with AD carries. I will not speculate on whether it's the best item build to have, I am only saying it is viable.