Without elaborating much there are three key points, with the first one giving reason for the second one to happen:
ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).
A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)
In dimension 4 (where the J-curves live) we have the adjunction formula.