They reveal the name of the plant hunter who discovered them, the place of origin, or a physical feature.
Some genera carry the names of people. The fuchsia plant is named after Leonard Fuchs, a 14th century botanist. Lonicera records Adam Lonitzer, the German 16th century botanist. Joseph Banks, the collector-botanist, is remembered through the banksia. Euphorbia was named after Euphorbus, physician to the king of Mauretania.
Atropa belladonna is the plant we commonly call deadly nightshade and the genus name carries a warning. Atropos was one of the three fates of legend whose speciality was snipping the thread of life.
Perhaps the best approach to botanical names is a recognition of what you see.
Look at the leaf shape. If it represents an open hand, palmatum describes it well. An almost round leaf would be rotundifolium, and lanceolatum must mean lance-like. Next comes leaf colour: viridis is green, aurea gold, and argentea silver. Rubens is red, caerulea blue and purpureus purple.
If hairs are visible, it can be described as hirtifolius. Hirsutus indicates they’re long, hirtiformis are bristle-like, and hirtellus means minute. This tells me that the cherry, Prunus subhirtella Autumnalis Rosea, presently blooming in the garden has tiny hairs under the leaves and pink flowers.
Several books deal with botanical names and terms, most following the lead of GF Zimmer’s dictionary, published in 1912. Some offer a guide to pronunciation, as in Plant Names Simplified by Johnson and Smith.
This said, there are no ancient Romans around to correct us.