The particular conjecture in question is about the power of $p$ defining the
Selmer group attached to the adjoint of the $p$-adic Tate module of an elliptic curve. (In general, the Bloch--Kato conjecture deals with the order and/or rank of Selmer groups.)
This order is supposed to be equal to the algebraic part of a particular special value of the symmetric square $L$-function of the elliptic curve. (Note that the symmetric square and the adjoint agree up to a twist, and while it would be more logical to speak of the adjoint $L$-function at this point, the symmetric square $L$-function is more traditional; in any case, these $L$-functions would be the same up to the change of variables $s \mapsto s+1$.)
Now results of Hida and Ribet show that up to small prime factors, this algebraic part of the special value in question coincides with the congruence modulus of the modular form attached to the elliptic curve. On the other hand,
Wiles's numerical criterion (or rather, the fact that it holds for the Hecke algebra in question) shows that the order of the adjoint Selmer group also
coincides with this congruence modulus. Putting these two statements together gives the case of Bloch--Kato under consideration.
(Diamond--Flach--Guo have a paper carefully detailing all this.)