In the book The Cosmic Perspective, it is stated that as matter is falling into a supermassive black hole, up to $40\%$ of its mass are converted to thermal energy, making the accretion of matter around a black hole a vastly more efficient energy source than even fusion. But what is the mechanism behind this conversion of mass to thermal energy?
The "mass" falling in is the rest mass (at infinity). As the matter falls it gains kinetic energy. Most of the matter cannot fall directly into the black hole because it encounters a potential barrier due to its angular momentum with respect to the black hole, so it enters some sort of orbit. The orbiting matter accumulates in an accretion disk.
To fall into the black hole, the matter must lose angular momentum. It does this via friction. At a microscopic level, interactions between particles and possibly with magnetic fields, transfer angular momentum outwards and also heat the disk material. The hot disk effectively radiates away (some of) the kinetic energy that the matter gained by falling towards the black hole and the matter moves inwards, eventually falling into the black hole. This radiated energy can be a significant fraction of the rest mass energy of the matter because it is moving relativistically when it gets close to the black hole.
One way of looking at the whole process is in terms of conservation of mass/energy. The start point is the black hole plus the rest mass energy of the material that is to be accreted. After accretion the black hole has accreted some of that mass-energy, but a fraction of it has been radiated away as it passes through the accretion disk. Thus, as @Sten points out, the mass accreted by the black hole is less than the rest mass that fell into it - the difference emerges as radiation.
All else being equal, the same object will produce less far-field gravitational force, the deeper it is in a gravitational well. This is not intuitive if you are used to newtonian gravity, but it’s the same principle as a helium atom weighing less than two deuterium atoms.
If gas is dropped straight into a black hole, this reduction is exactly cancelled out by the increase in far field gravitational pull caused by the increase in kinetic energy of the gas. If the gas is slowed down on the way down, then the effects don’t cancel.