Weird behaviour of a bath ventilator
You've wired the device up incorrectly. 

The three connections it is expecting are:

1) Permanent Live (always connected to live)

2) Switched Live (connected to the live side of the lights)

3) Neutral.

Not live, neutral and earth.

The fan uses the permanent live to power the motor, it looks at the switched live to decide when to do that, and it keeps the motor running for a while after the lights are switched off.

You should check the instructions as going on guess work is dangerous at this voltage, but from the pictures you've posted it looks like L is permanent live,  T is switched live and N is neutral. 

As it's wired up now the fan thinks the lights are off, so isn't running.

-- edit --

looking at the documents for the fan, this isn't explained well

https://www.hornbach.lu/data/shop/D04/001/780/498/785/73/8626717_Doc_06_DE_20160503161817.pdf#page=102

"An external switch e.g. light switch, thermostat, etc. must be used to control the device,  it  must  be  located  between  the  L  and  the  T wire."

If you're using this in a bathroom the usual layout would be something like the following....



(The terminal in the fan in the diagram called SL corresponds to your T terminal, and your local wires may be different colours)

The site from which that image is taken has a good explanation of how
to install a bathroom fan, covering the non electrical aspects as well.

http://www.ultimatehandyman.co.uk/how-to/electrics/install-shower-extractor-fan