Wifi boosters typically broadcast a network with the same SSID as the original access point. Your computers/devices will see them as a single network and "roam" between them, i.e. automatically switch to the stronger signal as you move around the house.
To verify that the booster is working, you can use a wifi analyzer app - most will be able to distinguish between the main AP and the booster via MAC address instead of SSID and display them separately. There are lots of such programs available, like this one for windows and this app for android.