Marx and Engels are referring here to what they describe as the internal contradictions of capitalism. Simply put, this means that capitalism contains the seeds of its own destruction. As it develops over time, the capitalist system will continue to create class antagonism between the bourgeoisie (those who own the means of production) and the proletariat (those who toil to produce the goods that the capitalist sells for profit). The development of capitalism, for Marx and Engels, inevitably means that it will become more exploitative.
The specialization of labor and innovations in plant, technology, and machinery will accelerate the drive for profit. In turn, this will lead to a growing impoverishment of the proletariat. As competition in the advanced capitalist economy becomes ever more fierce, capitalists will drive down wages, forcing a growing number of workers into wretched poverty. At the same time, the proletariat—under the guidance of intellectuals such as Marx and Engels—will come to develop a better awareness of what's in their best economic interests. Or, as Marx and Engels put it, they would develop a growing class consciousness, one radically at odds with that of their exploiters, the bourgeoisie.
Periodic crises are endemic to capitalism, but capitalism's innate contradictions will develop to such an extent that revolution will one day become inevitable. The proletariat will rise and the exploitative system of capitalism will be swept away. In continuing with their narrow pursuit of profit at the expense of all else, the capitalists are unwittingly undermining the very system that's made them rich and powerful. This is what Marx and Engels mean when they say that the bourgeoisie are acting as their own gravediggers.