One byte, one digit

Eight outputs, eight segments

The net list gives the map, and it is the tidy one: each output to the segment of the same position.

Nexperia calls the outputs Q0 to Q7; TI and most tutorials call them QA to QH. They are the same pins.

Why a 1 lights

The digit is common cathode: the eight LEDs share their negative end, and that end is on GND. An output that goes HIGH pushes current through its resistor and its segment to GND, and the segment lights. An output that is LOW leaves both ends of the LED at 0 V, and nothing flows.

So a byte is a picture of the digit. 2 needs a, b, g, e and d: bits 0, 1, 6, 4 and 3, which is 0b01011011, or 0x5B. The dot is bit 7 on its own: 0x80, added to any digit with | 0x80.

The table

const byte DIGITS[10] = {
  0x3F, 0x06, 0x5B, 0x4F, 0x66,   // 0 1 2 3 4
  0x6D, 0x7D, 0x07, 0x7F, 0x6F,   // 5 6 7 8 9
};

These are the same ten bytes as the 0.56-inch digit's common-cathode table, because that board's outputs reach the segments in the same order. A table from a common-anode tutorial has every bit flipped: 2 is 0xA4 there, and on this board 0xA4 lights everything but the 2.