how do you turn a placement into a career?

I am just starting my masters degree and part of the degree involves a work placement in the field. The course organiser has stated that these placements are the most invaluable part of our degree since over half of the students who undergo them manage to turn them into careers with their placement company. She also told several anecdotes about getting in touch with the companies to plan placements and finding out the person she was organising them with was a student 15+ years ago who just stayed in their placement and formed their career. I got placed into my dream company and the department i will be working with its pretty much a combination of all of my interests so im very excited about the work. The issue is its an enormous company. They get thousands of job applications a year and they are very in demand for their field and I don't know how good my odds are of turning that into a career since competition is so high. I would love to have a job with this company after graduation and I want to use this experience to try and achieve that. I tried looking online but a lot of the advice is very basic and I would do most of it anyway, such as dress appropriately, ask questions, be excited and show enthusiasm, listen and pay attention, learn peoples names and make contacts etc. I have done work experience before (all following the advice online about presenting yourself well to potential employers) but its never turned into a job. I just dont know how to take it up a notch to that level and need advice. Especially since this is my masters and my last real chance for student work placements before going into the graduate job market.