“I swear that I will pay true allegiance to Your Majesty, and to your heirs and successors according to law. So help me God.”
⬆︎is the swearing.
The most pertinent words are "according to law".
Parliament makes the laws in the UK, not the monarch, and if the monarchy ever got troublesome Parliament would change the law as it pertains to this king or any other.
As it did in the past. I seem to remember from history lessons that previous Charleses got sorted out good and proper for being troublesome and overweening.
Or, as minibaggs said, it basically means you promise not to join ¡s¡s (i.e. be a traitor to your country).
So, all in all, it's not a big deal.