It's possible to interpret Nige's statement - as he does - as "I'm only trying to help ..." - but anyone who knows anything about politics and politicians knows that the basic premise to any behaviour by any politician is either to obtain, or if obtained, to increase - power.
That is what politics is all about, and always will be.
So Nige is laughing because his bessie mate is now Numero Uno, and Nige can cock a snook at the government, and say "You're not laughing at me now are you, not when you need me, hah hah!!!"
Mrs May's reaction translates as - "Nigel, I would eat every one of my wonderful shoes before I would admit that I need you for anything other than occasional casual amusement on a bad day."
Which is pretty much what he probably expected, still, it passes the time.