How does a bathroom scale automatically power on when I step on it?
The scale is never really "off". Instead, it just turns off the display and other peripheral circuits such as the ADC and goes into a low-power mode internally. Every now and then (most likely based on a timer interrupt), it powers up the ADC briefly and checks the sensor to see if the weight has changed, and if so, it turns the rest of the peripherals, including the display, back on.

In low-power "sleep" modes, many microprocessors (and especially the MSP430 series) have a level of current consumption that compares favorably with the self-discharge rate of the battery.