The 8 pin IC is a real time clock, from the part number it's probably a Dallas part (now Maxim I believe). This maintains the time (notice there is a battery to power it) using a crystal to provide a known frequency which is divided internally to drive the internal clock.
The microcontroller communicates with this using a 3 wire serial interface, so periodically, the microcontroller will ask for the time and then drive the display through multiplexing as
@davesparks has explained.
Without knowing exactly what the firmware is in the microcontroller we can only guess at the exact function. But looking at it, R16 is a light dependent resistor, possibly to dim the display when the room is dark (this would be done by changing the duty cycle on the multiplexing for the display, so instead of each display being on for say 0.01s, they are only on for 0.005s and thus appear darker), R17 is a thermistor, so maybe the clock has a temperature function. These are connected as voltage dividers between 5v and gnd and would be sampled using the analogue to digital converter in the microcontroller.
P1 looks like it might be the programming connector to flash the microcontroller with the firmware.
S1 and S2 are the function switches. When they are pressed, the pull the input pin to ground (0), so they have internal pull up resistors in the microcontroller. These would be read by the firmware which would take the relevant action.
Beyond that, it's really a deep dive into the datasheets of the devices to understand what each pin is capable of and how that ties up with what it's connected to.
So for example....
View attachment 56632
U2 pin 4 is labelled as P1.1/ADC1, which suggests to me that it can be bit 1 of digital IO port 1 and the input for analogue to digital converter 1. How it's connected suggests it can also be the data line for a 2/3 wire serial interface as it's connected to the I/O pin of the clock. So this will be toggled high (5v) and low (0v) in sync with the clock pin (SCLK) during data exchange between the two chips.
To be able to fully understand this, you're going to need to read and understand the datasheets for the devices.