Electronic gears, how do they work?

There are three major component manufacturers: Shimano, SRAM, and Campagnolo. Each has their own electronic shifting system.

• SRAM's is simplest to recharge: the batteries are attached directly to the front and rear derailleurs; to recharge, detach one and put it in its dedicated charging base.
• Shimano has a "Di2 junction box" either underneath the stem or in one of the handlebar endcaps, which can be recharged with a USB cable.
• Campagnolo has a system called EPS that is similar to Shimano's.

I'll deal with the convenience of electronic drivetrains. SRAM's eTap groups have the shortest total battery life. In these groups, the shifters use a short-range wireless protocol to send shifts to the derailers. As Adam mentioned, each derailer has its own battery. SRAM claims the current AXS versions last up to 60 hours of riding time on a full charge. This is a fair amount of riding time. You could charge your batteries weekly. The CR2032 batteries in the shifters should last about a year.

Shimano's Di2 system has a battery in the seatpost, and wires connect the shifters to the derailers and the battery. This is harder to set up than SRAM. However, the battery lasts longer. A full charge can last months. I believe Campagnolo EPS has similar battery life. I have not heard about FSA's WE group, but uses wireless communication to the central processor, and wires between that, the battery, and both derailers. I'd expect similar life compared to Shimano and Campagnolo, plus you probably need to change the batteries yearly.

It's true that you now need to check battery life regularly. In partial compensation, you now don't need to deal with shift cables at all. I can assure you from personal experience that the batteries do last much longer than typical personal electronics, and other riders would say similar. They are not inconvenient once you get in the habit of periodic checks - although I suspect it can be harder to build intermittent habits than daily habits. You could set a reminder on your phone to check your batteries periodically.

In general, you can design lithium ion batteries to provide bursts of power or continuous power. I believe that electronic shifting probably uses power cells, since they would only need to produce short bursts of power to move the derailers (admittedly, they'd need to provide some continuous power if Bluetooth communication with a head unit is enabled). Our personal electronics most likely use energy cells. I've seen one page that rates the current Di2 battery at only 500 milliamp hours (i.e. it can produce 500 milliamps of current for one hour; this is a measure of stored electric energy). Each SRAM AXS battery is apparently rated at 300 mAH; I'd attribute the system's shorter battery life than Di2 to the fact that it needs to maintain wireless communication. In contrast, the current generation iPhone SE has a battery rated at 1,624 mAH, and most iPhones have larger batteries. Despite the apparently low capacity of electronic shifting batteries, they do in fact last a long time. Furthermore, the AXS batteries should charge within an hour, and a flat Di2 battery should get to a useful charge state within an hour also.