How can I show the admission board that I'm better than my academic record says
Abellán,

I grew up and attended college in Spain myself, then jumped to the US for a Ph.D. and been here for 20 years already. So I know the Spanish system well and have first-hard experience with the transition.

Your best bet is the recommendation letters. You really have to make sure that they come from people that know you well and that can speak at length and in depth about your qualities and potential. Letters from Spain don't always go to this level of depth; it would be best if your letter writers have had prior experience with this (of course not at the expense of their knowledge of you).

Letters are very important--the most consistently underrated component of an application by the students in my experience.

@lafemmecosmique's advice is very good. The best way to make your letters look outstanding is to do outstanding work with the people that will write them.

Good luck to you.