Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1) 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).

2) In dimension 4 (where the J-curves live) we have the adjunction formula.

3) 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.)

Here is another crucial point disguised as an application: On a symplectic 4-manifold with (negative) contact 3-manifold boundary, the standard "ECH curve count" yields a relative invariant in the $ECH_*$ of the boundary, while the standard "SFT curve count" yields a relative in variant in the (ordinary) contact homology $CH_*$ of the boundary. But if the contact form is *overtwisted* then $CH_*$ is necessarily trivial, whereas $ECH_*$ can easily be nontrivial.