The word coney (or coni, cony, etc) meant “rabbit”, and from there, presumably because rabbits aren’t the brightest animals, it came to mean “simpleton, naive person”. And in (thieves’) cant, coney-catchers were the rogues who scammed these persons, by cheating at cards or dice, or with various rackets and confidence tricks. So the coney, the rabbit, was the mark.
Alternatively, maybe the word coney-catcher came first, “by a metaphor, taken from those that rob warrens, and conie-grounds, using all means, sleights, and cunning to deceive them, as pitching of haies before their holes, fetching them in by tumblers, &c.”, and from there coney ended up meaning “mark” as well as “rabbit”.
Often a coney was someone from the country who just came to London, and wasn’t wise to the ways of the big city. Now that’s an easy mark. Rogues could spot them from a mile away, from how they dressed and talked and behaved (they must have looked so lost, imagine someone from a village crossing the London Bridge for the first time, an enormous bridge with a ton of buildings on it, and beyond that a sprawling cityscape: it must have been bewildering), and would immediately approach them and offer help, promise entertainment, etc. And then they’d fleece them.
So the rabbits in the picture, rolling dice and playing cards and getting hooked, are the victims that the rogues “caught”. And the top row, if I’m not mistaken, shows what’s in store for the rogues who get caught: the judge, the whip, and the gallows.