That sentence is the important clue you need. If you are still a junior programmer after 5+ years experience, there is probably something wrong. You should mostly ignore the job requirements in terms of the years of experience they want.
There are even plenty of inside jokes around this. Jobs asking for 10 years javascript experience a couple of years after javascript was released!
Either they are looking for a senior programmer and are hoping to pay junior programmer rates (possible) or they are just putting their list of ideal requirements (more likely).
Sometimes the requirements aren't even written by IT staff, HR writes the requirements, and HR doesn't have a clue about IT.
Unfortunately, this means that it's very difficult to filter for "true" entry level jobs.
Applying to jobs is mostly a numbers game. Unless the job makes it clear that you need to be a senior developer, or a team lead, then just apply. Before traveling to an interview, see if you can set up an initial phone interview. If the interviewer doesn't do a great job of screening you, ask questions yourself to try to understand your responsibilities better. Will you be working under a senior programmer? That's a good sign that they aren't looking for someone who can work 100% independently and that they are willing to mentor you.