6:28
" It may not do anything, in most cases it is not going to cause a problem at all. Unless there is some, you know, your working at audio circuits and there is hum and all sorts of things ...".
What he is saying is that if you connect the scope earth to a sensitive, high impedance part of the circuit under test it will behave like an antennae and inject some signal - probably main's hum - into your circuit. (It is actually your floating circuit that is the antennae, but the effect is that same.)