The display gives you key information while you are riding. It may be a dedicated display or an app on your phone. It may show your assistance mode, speed, range, battery level, trip distance, an odometer or a clock. The information available depends on what display you have. Higher spec models will have a better display and more info. GPS and Mapping is available on some models too.
If you have a remote control, it allows you to select different levels of assistance and cycle through the information available on the display at the touch of a button. No throttles here.
The battery is charged on or off the bike and acts like your mobile phone. Plug it in, it charges and you disconnect it when it is full. The battery is just a fuel tank, so the bigger the battery you have the longer the range is.
As you pedal, sensors in the motor determine your speed, how hard you are pressing on the pedals and the assistance mode you are in and works out how much assistance to give you. You are assisted upto a maximum 15.5mph. Beyond this speed the assistance stops until you slow down enough for it to kick back in again. Higher torque (Nm) motors perform better on hills.
Download a Bosch motor comparison here
Download a Yamaha motor comparison here