There's a little truth to what he says.

If a chain and cassette/freewheel are gunked up the gunk/dirt/grease/grime/oil takes up a tick small amount of space between the link/rollers/pins. Think of all that stuff as being like a 'shim' Remove all that and something has to take up that 'space'. Chain 'stretch' is what does.

Try it sometime. Measure a gunked up chain, clean it and measure again. That being said, you'd need an already stretched out and gunked chain for it to make a difference.