When you have a really old car, you are never sure that it is going to start. So, you let it run when you stop somewhere so that you will only get stuck at work or home if it dies out on you. Also, starters and alternators are prone to go out on older vehicles. Fewer starts extends the life of those components.
There isn't much difference in the amount of fuel you use now that all cars have very efficient fuel injection systems. Diesel trucks do lose a bit of efficiency if they are shut down and cool off instead of letting them idle. Most large cities have diesel idling rules that prevent the UPS guy from leaving the truck on. Diesels spit out more tailpipe emissions when they are idling because the engine is not hot enough to burn off all the NOx and SOx.
The remote starter is to keep from getting car jacked. Car jackers get you when you sit down and fumble with your keys to get the vehicle started. If the vehicle is running by the time you open the door, you can just slam it into gear, hit the gas and peel out of there. Also, tote the note lots like to trick out cars with alarms and remote starters to divert peoples attention from the fact that they have also installed GPS and a remote engine disabler in case you don't pay your note on time.