SD card breakout board
Why is a digital buffer needed in case? I didn't really get in general what digital buffers are used for.

Good question.
A buffer simply provides an amplification. Ask yourself: What happens if the output impedance of the sender's pin is not low enough? Or, what happens if the input impedance of the receiver's pin is not high enough?
In such cases, a buffer eliminates the potential problems with its very high input and very low output impedances.
Additionally (after seeing tlfong01's comment), HC4050 acts as a hi-to-low logic-level translator/converter. This means, as can be seen from the datasheet, that the IC can convert high voltage (up to 15V) logic level to low voltage (up to Vcc, mostly 3V or 5V) levels. This allows the user to use the breakout board with 5V-logic devices (e.g. older 8051s). That's why the breakout board has a 5V input and a 3V linear regulator (The regulator's output goes directly to the 4050).

Alternatively, would the second board be enough?

Could be, unless you keep the traces long. Remember that SPI is suitable for the devices sharing the same board. And also SPI does not like long traces as the speed increases.