Fins were on cars because fins were on rockets, and since rockets were fast, fins clearly needed to be on anything that moved fast, which meant cars. Pull back from that close-up and you'll see the rocket motor just below the fin on the Caddy. Our '58 Ford Country Sedan had much bigger rockets, but smaller fins.
I came across this link, which nicely tells the story of aerodynamic improvements through 1957, at least at Chrysler, culminating in a nice description of the crosswind issue.
And one that's more far reaching in scope.