In these paragraphs, Berkeley is anticipating Hume's arguments about causation, though ultimately, the overall conclusions Hume would reach a few decades later can be considered practically the opposite of Berkeley's.
Berkeley's point appears to be that when "natural philosophy" establishes a principle such as that of the attraction of bodies (in other words, gravitational force, as determined by Newton), the principle is really nothing more than an observation, extended from what is directly observable to that which is not; it is inferred from what we do see. On earth, we see gravity at work, and rather than restrict this property, the natural philosopher extrapolates it through analogy into a realm where it cannot be directly observed, as in the attraction among the bodies in the solar system.
Berkeley's argument is very much like that of Hume in asserting that when we observe two events in conjunction—such as a billiard ball striking another, followed by the motion of the second ball—we are assuming that these events, or similar ones, will always occur in conjunction, in sequence like this so that we conclude the first event "causes" the other. But both philosophers are stating that the human mind, in its tendency to draw analogies, assumes connections among different observable happenings and then claims the existence of a "law" that governs them, when in fact, no such law is observable. It's the essence of the empirical philosophy of both Berkekey and Hume that what is not directly experienced does not literally exist and needs some other factor to justify it or to establish it as provable.
For Berkeley, that "other factor" is God. The essence of things, the unobserved quality that establishes these analogized consistencies of "natural law," is the fact of their existence in the mind of the Deity. By contrast, although he starts from the same premise as Berkeley, for Hume, there is nothing. Beyond that which is directly observable at a given moment, nothing can be "proved." We see events and analogies, but beyond the immediacy of our perceptions, we have no proof that the principle of "cause and effect" exists and hence no proof, in Hume's view, that God exists. The two greatest empricists of the eighteenth century, Berkekey and Hume, start at the same point and reach opposite conclusions—the first in favor of religion and the second against it.