This work is over a century old, so a lot of the terminology is different. Many of the thermodynamic laws which are now taken as standard knowledge, including the First Law and the Ideal Gas Law, are not "givens" to Clausius or his audience. In particular, Clausius does not have the concept of "energy" to draw on so he is operating under the assumption that there is this fluid-like thing called "heat." He uses the "quantity of heat" to refer to the total amount of this fluid, and the "quantity of heat consumed" to refer to the net reduction in the amount of that fluid as a consequence of the process. If he assumes that the quantity of heat is constant, then it follows that the quantity of heat consumed is zero.
In modern terminology, we would associate the "quantity of heat" with the total internal energy of the two thermal reservoirs powering the heat engine, and the "quantity of heat consumed" with the net reduction in thermal energy of the reservoirs (which we know is equivalent to the work extracted from the heat engine).
Here are the passages which helped me understand Clausius' starting point and terminology:
In the steam engine, for example, by means of the steam which is developed in the boiler and precipitated in the condenser, heat is transferred from the grate to the condenser. He [Carnot] says expressly that no heat is lost in the process, but that the quantity of heat remains unchanged. .... If it be assumed that heat, like a substance, cannot diminish in quantity, it must also be assumed that it cannot increase. It is, however, almost impossible to explain the heat produced by friction except as an increase in the quantity of heat. The careful investigations of Juoule, in which heat is produced in several different ways by the application of mechanical work, have almost certainly proved not only the possibility of increasing the quantity of heat in any circumstances but also the law that the quantity of heat developed is proportional to the work expended in the operation. To this it must be added that other facts have lately become known which support the view, that heat is not a substance, but consists in a motion of the least parts of bodies. If this view is correct, it is admissible to apply to heat the general mechanical principle that a motion may be transformed into work, and in such a manner that the loss of vis viva is proportional to the work accomplished.
These facts, with which Carnot also was well acquainted, and the importance of which he has expressly recognized, almost compel us to accept the equivalence between heat and work, on the modified hypothesis that the accomplishment of work requires not merely a change in the distribution of heat, but also an actual consumption of heat, and that, conversely, heat can be developed again by the expenditure of work.
We shall consider here the kind of motion which can be conceived as taking place within bodies, further than to assume in general that the particles of motions are in motion, and that their heat is the measure of their vis viva, or rather still more generally, we shall only lay down a principle conditioned by that assumpion as a fundamental principle, in words: In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and, conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.