With moje ime je … you introduce yourself: “My name is …”. In the video Ljubica uses her own name as the example.
The image: you point at yourself, and above you is the moon that everybody can see. On the moon you have painted a hedgehog in flour, and every time somebody touches it the flour falls off. Moon, hedgehog, flour and the “every time” give you moje ime je.
As you listen, notice that at normal speed the first je almost disappears – it then sounds more like “moim ime je”. Say it slowly several times first, like practising the piano, and the speed will follow by itself.