How do aircraft rental companies charge while the plane is not flying?
Flying clubs and aircraft renters generally only charge for the time the plane is actually in use. In the U.S. at least, this is commonly measured using the Hobbs time (actual clock time used, triggered by oil pressure or some other indicator that the plane is "in use"), or by Tach time, which is a bit trickier but can very roughly be thought of as the amount of wear you've put on the engine. (Strictly speaking it's a count of the number of engine revolutions, converted to "hours at cruise RPM".)

For overnight or longer trips, it's common to find a minimum hourly usage per day. For example, if you take a plane on a 3-day trip, and your club/owner requires 2 hours/day, you will be paying for 2 x 3 = 6 hours of Tach / Hobbs / whatever time, even if you didn't actually use the plane that much. You'll pay for the higher of your actual usage or your minimum usage per day.

It's extremely rare in the U.S., but not unheard-of, to encounter an extra fixed fee per day for taking the airplane for more than one day. (I've never personally encountered a fixed fee like this.) I'm not sure if things are different in the U.K.

Hourly rates can also either be "wet" (the rate includes fuel: any amount you pay for fuel will be deducted from your rental fee when you return) or "dry" (the rate includes the airplane only and you must pay for fuel). "Dry" rentals may or may not also require you to pay for oil or other consumables used.

(This is general information. Of course, if your question is specifically about Fly ASG's policies and fee scales, your best bet is to contact them directly. I know nothing about them.)