A .me domain is all about you. It's a way to show the world what makes you the special ray of sunshine that your mother always said you were. And .uk is the domain for the United Kingdom of Great Britain and Northern Ireland. Put the two together, and you can tell citizens of the UK all about you and your world.
All jokes aside, you are different than anyone else. And if you live in or do business in the UK, then having a .me.uk domain shows the web your personal side. And once they learn how brilliant you are, that opens all sorts of doors.