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 "secretly" 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.
That is a green LED. To make it light up, it needs about 2 volts. You have two 1.5 volt cells in series as a battery, so you have about 3 volts total.
The two volts for the LED leaves only one volt for the motor. That one volt can't force enough current through the motor to make it turn.
The LED also has some resistance - it limits the amount of current that can get to the motor.
Both effects together prevent the motor from running.
Connect the fan and the LED in series, and give the fan a spin. The motor should try to keep it running. The LED will also change brightness when the fan spins. It should get dimmer when the motor is turning, though it may also "blink" brighter as the motor resistance changes due to commutation (switching of the active coils as it rotates.)
In a series connection all the current flows through all the parts.
The LED doesn't pass enough current to operate the motor that's behind the fan.