Hi Natasha, and welcome. Jay is absolutely right. Bromeliads do die after they bloom only once, but they make new babies to keep the family going. You have a nice healthy pup out the left side of yours in the picture.
Mama might make more pups before she is done, so don't do major surgery yet. You can cut out the dry flower stem with a pair of scissors, and remove any dead leaves, but keep any leaves on the mother plant as long as they are green.
Bromeliads are epiphytes, so they gather a lot of their moisture and nutrients from the leaves, and really not much from the roots. The roots are just to anchor the plant in place. So spraying the foliage is just as important as keeping the soil in the pot moist. I have orchids (which are also epiphytes) so I am often walking around with a sprayer, and at least weekly I spray very weak fertilizer on all the orchids and the bromeliads sometimes too. If you can get a little bit of soluble fertilizer and keep a spray bottle of water around that will make them very happy.
They really don't make a very big root system so re-potting isn't usually necessary unless you have so many new pups in one pot that they need more room to spread out. You could just wait until Mama is completely dead, then lift the whole thing, remove her and set the pups right back in where they were.
I've just thrown bromeliad pups on the ground here and had them root themselves and grow beautifully. Lots of shade, humidity and leafy ground, and they grow well.
to ATP. You have done the right thing to post your question.
Nothing much I can add to Elaine's answer. I just remove the leaves when they brown and yes, cut away that dead flower/stem. Generally the lower leaves will turn brown and brittle first and then the leaves progressively up to the crown of the plant. I might remove only a leaf of two at a time and it might take months before the "mother" plant has completely died.
