A cut dog has no pups. Don't cut cards.
The obvious reference that comes to mind is that a cut dog has been neutered, since he has no pups. However, I could find no explicit discussion of how cutting cards might put one at risk of castration.
Amarillo Slim Preston offers an exegesis of this admonition which may or may not help in your understanding: “Poker players shouldn't get into that bad habit of wanting to 'cut the cards, cut the cards, cut the cards.' People say that a cut dog has no pups, and that's the damn truth. That kind of cutting the cards every time you turn around is for people either hunting for luck, or trying to cut off something with their hands...”
Card games appear to be magnets for dog references. A bulldog is the king of any suit. The ace of clubs is called the puppy foot.
Preston , Amarillo Slim, and Bill G. Cox. 2005. Amarillo Slim's Play Poker to Win. 1st Collins ed. New York: Collins, 143.
Haber, Tom Burns. 1965. Canine Terms Applied to Human Beings and Human Events: Part II. American Speech 40 (4): 251.