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I'd appreciate it if anyone could explain to me why do we demand that the wavelength will be much smaller than the distances between the slits in order to see wave diffraction when performing Young's experiment.

From searching online I've understood that if $$ \lambda \gg d $$ where $d$ is the distance between two slits, then the maxima which occur every $\lambda/d$ will go out of the screen and we won't be able to see them. Is this correct?

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Without resorting to mathematics, the following picture should speak a thousand words:

Wave diffraction

As the slit size increases with respect to the wavelength, the effect of wave diffraction becomes smaller and smaller. So to observe diffraction the wavelength should be smaller than slit size.

Image credit.

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