This is actually not a completely settled question, so don't feel too bad if you don't understand it in every detail! It's also possible that the cause is not the same in all situations, and be aware that my explanation is not universally accepted, but here's my interpretation:
There are actually two interacting phenomena going on: the formation of the attached eddies (vortices), and the wake instability. The formation of the eddies occurs because the fluid flowing past the cylinder, but at some distance away, is moving fast relative to the fluid directly behind the cylinder. The shear (or, if you like, the low pressure behind the cylinder) redirects the flow in towards the center of the wake and ultimately (some of it) back towards the cylinder, creating the two symmetrical eddies, one on each side. This is a steady-state solution to the Navier-Stokes equations given that the flow be zero at the boundary of the cylinder - but it is not necessarily a stable solution.
Meanwhile, in the wake instability, for certain flow parameters the center line of the cylinder wake begins to wobble. The reason for this still under study, and it is unclear exactly what role the attached eddies play at the beginning, if any, but it is a well-known instability. (This is the controversial part - many attempted explanations of the vortex shedding instability ignore the wake instability, or assume that the wake instability is a consequence of the instability of the attached eddies and therefore that it can't be a cause of the vortex shedding. I think both of those lines of reasoning are erroneous, though there is certainly a mutual interaction between the attached eddies and the wake instability.)
As you increase the flow rate, two things happen: the attached eddies get longer, and the amplitude of the wake instability increases. Eventually the eddies are long enough and the wake instability large enough that the peaks in the lateral wake flow can pinch off the eddies. Because the wake instability is wobbling back and forth, it pinches the eddies off alternately on one side and then the other, and you get the vortex street.
It is known that the interaction between the vortex shedding and the mean wake is nonlinear, in the sense that the most unstable mode of the initial steady wake initiates the instability, but then as the vortices grow they alter the mean flow, which changes the most unstable mode, etc., until a saturation point is reached. That saturation point is what is usually observed as the vortex street.