I believe "absolver el traslado" means to answer the notice/summons.
In this case, although expressed in the present tense, the "demandante" has fulfilled the procedural step of answering the notice of, I assume, the opposing party's challenge or objection, by submitting the briefs dated December 2014 and April 2015.
You might like to add something like "as required" after it or word it as "the court acknowledges that the plaintiff/complainant has answered..." in order to include the sense of "cumple con".