Qualifications
This is a federal transportation safety position that gets paid a lot of taxpayer money, so the bar is rather high when it comes to who the government will consider for the job. First and foremost, they won't let anyone near this job unless they're a U.S. citizen who can pass a drug test.
They'll also need prior experience working on the railroad all the live-long day. Depending on your specialty, you'll need anywhere from one to five years under your belt before even thinking about a career in safety (source).
You'll also need to be able to obtain and hold a government contractor-issued travel card. This is the card that you flash around to railroad employees so they know you're the real deal, and you mean business. This includes regular credit checks and financial statements—that way, the government can make sure you're not falling into debt or accepting bribes from shady railroad executives and managers.
Proving your worth comes during the application and training processes. Besides the references and experience, you have to demonstrate that you didn't make up everything on your application (which, by the way, is a crime). As the FRA itself says, they're looking for people who can collect data, turn it into facts, and then figure out what it means and what to do with it.
They also want people who actually know trains intimately—so if your past experience solely involves watching trains on YouTube, that isn't going to cut it. Surprisingly, one thing that isn't required is a college degree, although it certainly couldn't hurt to have one. As long as you have the experience or training necessary to get the job done and can back it up, you'll do fine.