"One's agent is as oneself"
The Torah established a foundational principle: "one's agent is as oneself" (Kiddushin 41b; Rambam, Laws of Agents and Partners 1:6-7). When a person appoints an agent to perform a mitzvah, the agent's act is considered as though the sender performed it himself. This principle opens a door for Diaspora Jews: one who cannot stand himself in the field in the Land of Israel may appoint an agent to perform the act in his body, and the fulfillment is credited to his account.
Partnership through money
But agency alone is not enough. The Rambam (Agents and Partners) teaches that participating in the costs of the mitzvah turns a person into a partner in the act. The Midrash (Tanchuma, Terumah 9) states regarding the donation to the Tabernacle: "one who participates in building the Tabernacle is as if it were all built through him" - contributing money credits the giver as if he himself performed the act.
The "magic" of partnership
This is the "magic" of Kerem 613: the partnership in payment is not mere donation, but real participation in the cost of fulfilling the mitzvah. The emissaries in the field act with their bodies, and the partner pays the cost of the act. On the basis of "one's agent is as oneself" and partnership through money, the mitzvah is credited to the partner as if he performed it himself.
And so the Sages taught (Sotah 14a): "Whoever assists those who perform a mitzvah, Scripture considers it as if he performed it."