A catalytic flameless combustion apparatus has a fuel inlet, a combustion-supporting gas inlet, a gas premixer, a combustion plate, an igniter, a gas deflector, a flameless combustion cavity, a catalyst filled in the flameless combustion cavity, a gas collection chamber and an exhaust port. The method for starting the catalytic flameless combustion apparatus includes initially combusting and heating the flameless combustion cavity and the catalyst filled therein with low power flame; and then increasing flow velocity and switching to high power flame for conducting catalytic flameless combustion. The catalytic flameless combustion apparatus can be used for various non-solid fuel combustion and heat extraction processes.