On a machine for producing articles, in particular a packaging machine for producing packets of cigarettes, a reel, laid flat on a support, is fed, by means of a transfer device, to a relative unwinding pin fitted to the periphery of a rotary platform rotating about a substantially horizontal axis; the unwinding pin being set to a substantially radial position with respect to the platform axis; the rotary platform being rotated to set said unwinding pin to a vertical receiving position facing upwards; the reel being fitted downwards onto the unwinding pin in the receiving position; and the unwinding pin being rotated, with respect to the rotary platform, into an axial position with respect to the rotary platform before unwinding the reel.