How do I properly level convert between microcontroller and 3.3V SPI pins
I wrongly assumed that If I supplied to MAX31856 3V3 to the digital
  and analogue inputs of the chip then why would I need to level
  convert? Now I see that I need that for SPI lines.


You absolutely do NOT need that for SPI lines, since both STM32 and MAX31856 work at 3.3V.


  I'm assuming the diodes are acting as some sort of level shifter
  however I'm not entirely sure. Can someone explain the purpose of the
  diodes.


Diodes in combination with pull-up resistors work as uni-directional level shifters on Adafruit board, basically imitating open-collector operation of I2C bus and making it compatible with 3.3~5V MCUs. Again, you do not need this.

The operation itself is very simple. When MCU outputs 0 the diode opens and pulls MAX31856 input down to 0. When MCU outputs anything higher than 3.3V the diode closes and allows pull-up resistor to supply 3.3V ("1") to input pin. It does not allow 5V output from 5V MCU pass through to MAX31856. Note that this severely reduces the maximum possible data rate of the SPI communication due to relatively weak 10k pull-ups reducing slew of the rising edge.