Why doesn't the fan work when the LED is connected in series with it?
Figure 1. Extract from 288 Amazing Science Station user manual.
A little care has to be taken when using LEDs to limit the current to a safe value. Connecting, for example, a 2 V green LED across a 3 V battery may destroy it. Limiting the current through the LED is usually done by adding a series resistor to the LED. I suspect that the 288 Amazing Science Station has current limiting resistors &quot;secretly&quot; added in series with each of the LEDs on the station to prevent accidental destruction. For a 3 V supply this might be a 100 Ω resistor which will drop 1 V at 10 mA - plenty of current to make the LED glow brightly.
When you connect the LED in series with the motor the maximum current that can flow is 10 mA (or whatever the LED current is set at) and the resistance of the motor will reduce this a little. There just isn't enough current to run the motor.
Note that in Experiment 3 the fan and the LED are connected in parallel with the battery rather than in series as you have done. I can't make out what the additional component is but they seem to have added additional resistance to the LED for this circuit.