'cos that's what they all do! seriously, though, the flowers (the whole stem is called a raceme, I think) don't seem to last all that long. This year my wisterias weren't doing their best - due I suppose to the amount of rain we had when the flowers first began to open. Those at the top of the racemes were well past their best & starting to die off before the whole had flowered. What's usually left after flowering is a rather tough central "stem" which stays put even over the winter months, giving the plant a bit of a whiskery look sometimes. I just snip them off - takes hours as the wisterias cover most of the front of my house! If any of them set seed, the pod looks a bit like a tough hairy runner bean pod and this will stay on the vine for months if you let it. I don't think may people attempt to grow wisterias from seed - I believe they're mostly propagated using cuttings. In any case, they take years to grow & be mature enough to flower, so if you're considering buying another one any time I'd suggest you get one which is in flower at the time of purchase.
