If you are lucky, the sponsors come to you. They observe your buzz and will want to work together with you to promote their brand. Most of the time however it is the artist that will want to reach out to bigger companies and brands for sponsorship. Having that first meeting indicates that they are open to your proposition so the rest is up to you to convince them that you will be worth their while.
Before they even say yes to sitting down with you, try to delve in why you are approaching them in the first place. Is it simply for merchandise or financial support, or is it for something that's long term? Don't assume you know what they are looking for, so be genuine in your intentions. If you have a vision, make sure it is consistent and something that is true to your craft. Be determined but not overly confident and you will likely get them to warm up to you.
You might face rejection after knocking one or two doors but that's all part of how sponsorship works. Refine your proposal and have clear objectives in mind. There are plenty of companies that are just as hungry to work with talent and influencers so be sure to cast your net far and wide. You never know if your next deal might just be waiting around the corner.