The stop start feature is activated when the Eco button on the centre console is activated.
In essence when you stop at say a set of traffic lights, put the car in neutral and lift your foot off of the clutch the engine will shut down. On the Evoque the parking brake come on automatically as well. Thus cutting down on fuel useage and CO2 emissions.
In my own car this only happens when a. the engine has reached operating temperature, b. the battery has a decent charge and also you are not putting too much load on idle, i.e lights on, a/c on, fan on.
It is quite strange to get used to it, some people love it some hate it, but you can switch it off.