Sri Sri Ravi Shankar:
Sankalpa means taking our consciousness to the universe, to the infinite; and then bringing the mind to the present moment, and making a wish for something that you long for in your mind. That is what Sankalpa is.
In taking Sankalpa, you keep something and then you pray for something. All the Yagnas and Poojas start with the taking of the Sankalpa.
Yagna means that which is performed by everyone together. So everybody is a part of it. Whoever does any Seva naturally becomes a part of it.
When anyone does any service for the yagna, they become part of the yagna. Even if they are serving in the kitchen or taking care of the devotees, then also they become a part of the yagna.
But if you have some wish for somebody in your mind; a friend or family member, and if you want some of the merits (of the yagna) to go to them, then you can take a Sankalpa.
When you take the Sankalpa, you take your name and mentally remember your wish, or the person for whom you want some merit, and then there are little rituals that they (the priests) do for you. So that is also possible.
This custom has been there for many years. It is not that someone will get a special benefit by doing this. Everyone who participates in the Yagna and has faith will be benefited. But if you want to do it with a special Sankalpa, a provision for that is also given.