You don't need the link to answer the question, it just add context.
3 LED and 3 buttons means 7 wires... is there a way to do it with fewer wires?
A keyboard does it with 4 wires... I don't know how... would that idea work?
If yes, can someone tell what to google, I don't know the words?
If not, what would?
Yes, it is possible.
There are several ways to do it.
I will list at least 2 ways.
For example, several (6 or more) buttons can be "interconnected" with a controller with only 2 wires. Using analog ports.
You can also interconnect buttons, LEDs, etc. with only 2 wires, using 1-Wire technology.
Make a drawing (or sketch) of your project, even freehand, so that we can better understand your needs and help you better.
You could use a PFC8574 chip. This will require 4 wires.
But for the best advice, you are going to have to give us more details of the rest of your circuit. How does the screen connect to the SBC for example? Many use SPI interface. In that case you could use a MCP23S08 chip which will require only 1 more wire because it can share other wires with the display.