Scale invariance of a theory means that if we notice one phenomenon on one scale (like an electromagnetic wave with the wavelength 400 nm), we can expect the similar phenomena to be possible at any other scale: 400 m, 400 km, etc. The magnetic field of a small magnet has exactly the same shape as the magnetic field of a larger magnet of the same shape, just rescaled. It should only be remembered that all relevant qualities are appropraitely rescaled. EM is rescaled under the transformation $x \mapsto b x$, $t\mapsto b t$, which means that when you rescale spacial dimensions, you also need to rescale time appropriately. That means that that if you consider a wave that is $b$ times shorter in space, it will also have $b$ times shorter period of oscilations in time, etc. So you not only know about the existence of waves of other length, but you can also predict their properties.
The lack of scale invariance means that observing one phenomenon does not guarantee the existance of similar phenomena at other scales. For example, all atoms have similar size (which can be calculated from the mass and charge of the electron, and there are no meter- or kilometer-big atoms. Another phenomenon is connected to the annihilation of elctrons with antielectrons: the photons created in the process have fixed energies (and wavelenngths). There are also more subtle effects, for example, through the creation of virtual electron-antielectron pairs, photons are able to interact with each other, and the strength of this interaction depends on their energy/wavelength.