The HDMI port is a small connector soldered to the edge of the mainboard. It carries every video signal the console produces, and it is the part most exposed to physical force. Trip over the cable, or move the console while it is plugged in, and the port bends before anything else gives way.
Because the damage is confined to that connector, the rest of the console is usually untouched. Replacing the port restores the console to full working order, with your storage, games and saved data exactly as you left them.
The work itself is micro-soldering: the old port is removed with hot air, the pads are cleaned and inspected for lifted traces, and a new port is fitted and tested. It is precise work, which is why most shops send these consoles away rather than repair them.