Why do anti-trust regulators in China need to approve NXP's takeover?
Generally, antitrust regulators have authority over any company doing business inside the zone of its control, hence why the EU antitrust regulators hit Google in 2017, or the EU's recent examination of US company Apple's attempted acquisition of another US based technology firm, Shazam.

Of course, for these companies, there's always the option of simply pulling out of those markets (not selling any goods or services, or holding any capital inside their borders) which dodges their authority. But given the size of these markets (US, China and the EU particularly), that's simply not a feasible option.