"Drive" in this sense means a private road connecting the main building on a property to the public road. It usually is used with reference to a road leading to a large structure, eg a mansion, through a scenic or decorated landscape on a large piece of property. "Sweeping" here implies "gracefully impressive" and easily traveled. Many such drives end in a carriage sweep that allowed horse drawn carriages a graceful and easy way to change direction and exit after entering.
EDIT: As always in defining words or phrases, I should specify that I am not qualified to discuss nuances in British English. I tried in my topic sentence to indicate the denotative meaning being used: a private road joining the public road and the main building. In the US, such a road if short may occasionally be referred to as a "drive" but will typically be referred to as a "drive way" with "drive" being reserved for an unusually long such road. Apparently usage is different in the UK. However, in the passage quoted by the OP, the idea of a substantial road in a pleasing landscape on private property is what is meant, not a slab of concrete 2 meters wide and 3 meters long.