The line is spoken by Leslie Nielsen's character, Professor Fate, who is a rival racer and antagonist to Tony Curtis's character, Leslie. It's used in a humorous way to describe a way to cheat in a race.
Here's the scene where the phrase is used:
* Professor Fate: "Listen, there's a way to do this thing so that we both win. It's a simple matter of a few subtle adjustments to my car. Roman hands and Russian fingers. You catch on?"
* Leslie: "Roman hands and Russian fingers?"
* Professor Fate: "That's right! Roman hands, they know how to make 'em nice, but they're not much for speed. Russian fingers...well they know how to go fast, but they're not so good at the details. You know, like brakes."
Professor Fate then proceeds to sabotage Leslie's car, causing it to malfunction and making him "lose" the race.