What nomenclature do names like fumarate and malate derive from?
Unfortunately, you will have to memorize the structures and the names because there is no chemical structure hidden in those names.
From Elsevier's Dictionary of Chemoetymology

fumaric acid $\ce{C4H4O4}$, derived from the genus name Fumaria (fumitory), from fumus (Latin:
smoke)

Fumaria plant


malic acid $\ce{C4H6O5}$, derived from malum (Latin: apple)

The same goes for all other common names like oxalic acid, acetic acid etc.