Solution to a Np-hard problem and its relevance to a dual LP
Any linear program has a dual linear program, whose constraints are in one-to-one correspondence with the variables of the primal program. Under mild conditions, the optimal value of the dual program coincides with the optimal value of the primal program, and certain conditions known as complementary slackness are satisfied by any pair of optimal solutions.

In this case, however, we are already looking at the dual linear program; the constraints here correspond to subsets. In detail, the constraint $\sum_{i\colon e_i \in S_j} y_i \leq w_j$ corresponds to the subset $S_j$.

What is the relation to Set Cover? There is a natural linear program which is a relaxation of a natural integer programming formulation of Set Cover. If you compute its dual, you get the linear program that you quote. (If you're not sure which primal linear program was used to generate this dual linear program, you can always compute the dual of the dual linear program to get back the primal linear program.)