The flowers just fall off and leave a dead end of stalk That's because they're male flowers - at least the one in your picture is. They tend to produce several male flowers before producing their first female flower, which you'll recognise because it'll have a small courgette, squash or pumpkin behind it. If it's pollinated, then the flower will stay on for a day or two and the small courgette etc will start to grow.
If you look closely, your male flower will contain just anthers giving ogg lots of pollen, and your female flower, when it arrives, will have a sticky stigma and no pollen.
If you don't have any female flowers needing pollinating, then the male flowers taste delicious, very sweet. Either shred them on to a salad, or google recipes for courgette flowers in batter or stuffed courgette flowers.
Your plant could do with some fertiliser - the leaves are looking a bit pale.