You’ll need to prove your status by supplying a .edu email address, but you will get a 6-month free trial if you’re new to Prime. And yes, the $49 yearly rate remains intact if you’re confident you’ll use Prime for long enough in a given year.
It’s not at all surprising that Amazon would cut students another break. Amazon might make little to no money from Prime if you’re a per-month customer, but there’s a real chance you’ll order enough products to offset that difference. And yes, Amazon is likely hoping to snag some long-term customers. Even if you’re not a profitable subscriber while attending school, Amazon no doubt hopes that you’ll like Prime enough to pay full price once you graduate.