Wow, an actual computer with flashing lights. I wish computers had more flashing lights. Isn't it the future yet?
The lights are on the input/output boards mostly; they indicate if an input or out put in on or off. Someone familiar with this system would be able to spot an issue in the system with one look (one stuck valve that stays open or closed when it should be in the opposite state would have a light on or off when it shouldn't).
Modern PLC doesn't feature LEDS or more rarely since their processing power improved and the informations are displayed on color LCD display, or to a computer, or they send alerts on SMS/phone/email.