I'm not sure why they call the R part of those carboxylic acid examples a "Radical", it's just an alkyl group.
A possibility maybe that they consider it a radical since the carbon is donating only one electron to the bond from it's valence shell and that the R group is only donating one as well. So if you treat them seperatley only posessing the electron that is originally theirs, I guess you could refer to them as radicals.
But it's a stretch and is pretty misleading.
Letting R stand for "Rest" makes a hell of alot more sense. Since that is how it's used.
And yeah that OP structure represents a hydrogen bonding formation found in a type of protein secondary structure called an "Anti-paralle Beta sheet", with the R groups represesnting amino acid side chain.