While the US and the Soviets had their own distinctive tech, it was still built on the basics of science. There was also no shortage of copycatting, espionage, etc. Basically, you copy the leader. They have tech that works, so why reinvent the wheel? So you wind up with an amalgam of tech, gleaned from the wild, and all of it working mostly the same way. It's not unrealistic to assume that a US pilot could fly a Russian jet (albeit not as proficiently), since all planes still have to work in the same basic way.
It stands to reason that the Cardassians (and indeed most of the Alpha quadrant) worked mostly the same way with warp drives, transporters, sensors, etc. So as long as you knew how the device was supposed to work, it stands to reason that you could make it work (provided the key components worked or were similar enough to repair). So, while Chief O'Brien didn't know how the Cardassian transporters were built, you could presume the interface would be similar enough to make an educated guess.
It also stands to reason that the Cardssians probably directly stole some of that know-how from the Federation. Seska, the Cardassian mole in Voyager, proves the Cardassians have their own infiltration system. It's not unreasonable to assume they would be looking at near-copies of their own panels, just labeled in a different language. We saw something similar in Star Trek III, where the crew has taken over a Klingon Bird of Prey
256 BRIDGE - BIRD OF PREY 256
Kirk in command, Sulu and Saavik on nearby station.
They are confronted by familiar tools, but baffling zymology.
KIRK Anyone here read Klingon?
No one does.
KIRK (continuing) Well, take your best shot.
257 ENGINEERING PANEL 257
Scotty hunched over the console, Chekov hunched over him...
CHEKOV If you can bypass into the module -
SCOTTY Fine, but where's the damn anti-matter inducer?
CHEKOV This.? No, this!
SCOTTY This or nothing.
Scotty makes several adjustments, presses some buttons,
then, with a deep breath, moves a sliding handle for-
ward. A HIGH WHINE STARTS, WAVERS, THEN GROWS ROBUST.