Why is the elevation of the Iberian Peninsula so high?
I don't know for sure, but my guess is that at least a part of the high Iberian elevation is due to NNE-SSW compression in northeast Spain, caused by rotation of the Iberian block. That is,the same rotation that opened up the Bay of Biscay deep basin, whilst compressing the Pyrennean mountains. The gentle dip of most of Iberia towards the southwest would fit well with such an explanation.
But then you may ask: 'Why is there a low elevation sedimentary basin just southwest of the Pyrenees?' Check the age of sediments in this trough. I would bet they are pretty recent, and I suspect that there may well be some late-stage counter-rotation caused by the  relative east-west displacement along the Rif-Betic arc.