Step One: Fight Ignorance.
It is not a myth. At least, not technically, in that a myth is a form of folk narrative having to do with gods, heroes, and the like. Comparatively few myths involve cats. What we have here is a folk belief. I understand the use of “myth” as a term for “folk belief which also happens to be untrue,” but in the academic field that studies such things (folklore) myth is used for something else entirely.
Step Two: Answer Question.
The belief is part of a general complex of beliefs about cats being unlucky, which probably has to do with their quasi-social, quasi-solitary nature. The sucking breath belief is attested from 1607, with skepticism about it attested from 1708. It is also unlucky for a child and kitten to be born on the same day, for a cat to climb over a pregnant woman (across her lap, presumably). Lady Wilde quotes an Irish greeting, “God save all here, except the cat.” On the other hand, since folklore is nothing but contrary, there are a number of folk cures involving cats, and beliefs that they are lucky as well as unlucky. (Sometimes this varies by region: black cats are lucky in New England, unlucky in the Midwest).
As far as the ultimate origin, it is simply impossible to say. Katherine Briggs wrote the book on cat folklore (Nine Lives), and she mentions the belief but doesn’t attempt to explain it. There have been some good suggestions in this thread based on cat behavior, but it is such an old belief that it is probably impossible to give a factual answer. The questions to ask are:
Q. Why cats?
A. Probably because their penchant for soft, warm places and (sometimes) sleeping around people’s heads led them to hang out by babies’ faces; also the milk suggestion above.
Q. Why sucking the breath out?
A. There is an old belief that the breath = the soul (in Latin, words for spirit and breathing are the same, and both are present in earlier senses of the English word ‘ghost’), so if you want to kill someone magically, sucking out their breath is like sucking out their soul.
Q. Why would a cat harm a kid?
A. Cats are associated with witches, who exist just to harm; this association is at least as old as the attestations of this belief.
A#2. Folk tradition needs an explanation for infant deaths; blaming the cat, a disposable member of the pre-modern household, could have psychological benefits that accidents of fate or poorly-understood diseases would not have.