How to extract energy from an alternating electric field?
Given an alternating electric field E inside e.g. a big parallel
  plates capacitor


Assuming 1 sq metre plates and 1 metre between the plates, there's a total parallel-plate capacitance of 8.85 pF. Try this calculator to check.

At 50 Hz the capacitive reactance is 360 Mohms so how much energy do you think you might get from two smaller plates on the inside? The smaller plates might be 0.1 sq metres and then the capacitance from an outer plate to a smaller inner plate will be 0.89 pF having a reactance of 3.6 Gohm. But there's two of them so that's a total series capacitive reactance of about 7 Gohms.

What were you expecting to see across your meter and were you in fact thinking that a capacitive transformer would result?

If the smaller inner plates were each 1 mm from their respective outer plates then the total capacitive reactance would be 7.2 Mohms and you would certainly see some voltage on your meter.