The move to smaller engines is not to help the slower teams. It really won't help them much at all since they don't develop their own engines. Right now the only engine suppliers are Ferrari, Renault, Mercedes, and Cosworth.
The reason for the switch was two fold. First, the FIA is always looking for a way to reduce costs. Lowering costs means more teams can compete. The original idea was to use a 1.6L I4T because the FIA has mandated I4s in all their other sanctioned racing series (WRC, Touring Car, etc). So the idea was that engine developers could use data from the other forms of racing and use it to cross-develop the engines, cutting costs.
The second reason was specifically to make F1 more relevant to street car development. Small displacement I4T engines are where the world is headed as far as everyday cars go. The switch was supposed to make it easier for F1 tech to be applied to road cars, making the car you buy more powerful while also being more efficient.
Unfortunately, the engine manufacturers couldn't agree on the I4 thing. Renault wanted it and had it as part of their contract. Cosworth, being the smallest of the manufacturers, will go the way the wind blows. But both Ferrari and Mercedes did n't like being limited to only 4 cylinders. Renault and Cosworth could not stand up to them, but F1 can't afford to lose two engine manufacturers. So a compromise was struck and they agreed on a V6.