This statement is entirely applicable to the story, even if we do not necessarily accept the idea of fate as an active supernatural agency as represented by the monkey’s paw. Indeed, it is suggested within the story itself that events may be put down to 'coincidence' rather than the magic of the paw. What is unquestionable, however, is that the Whites wish for more than they already have and disaster ensues. They want more money and they get it – but only at the cost of losing their son.
The Whites' punishment is unduly harsh, but they do appear at fault in wishing for the money. The clue is in what Mr White says when he is first about to make a wish:
"I don't know what to wish for, and that's a fact," he said, slowly. "It seems to me I've got all I want."
The Whites, then, do not really need this extra money. They are already quite comfortable and happy. This wish, therefore, represents greed rather than an actual need.
The Whites, then, show that they are not content with their lot, they attempt to exceed what has been ordained to them, and they are punished for their hubris. It seems they upset the natural order of things and everything goes haywire for them. This is the sense in which they ‘interfere’ with fate.