How do I research what a potential sponsor company actually wants before approaching them?

Study the company and their business goals thoroughly before presenting. The good news is that companies are often willing to share their goals if you simply ask. Look at where they're shifting their marketing focus (TV, print, social media), what new products they're launching, and what new markets they're entering. Understanding their business objectives allows you to position yourself as someone who can help achieve those goals, rather than just asking for money.