There are multiple angles to sponsorship. It can be partially tracked by media, sales, and the sponsor's own platforms. But the process is like the social sciences and not hard sciences.

Sample scenario:

In February 2016, my peripherals company starts sponsoring an EU LCS team (let's call it Team Zero).

We announce it on our website, through our social media, and on press releases to media. Nobody really cares that much, but we see an increase of likes, shares and a few more emails.

In March 2016, we go to a tech and gaming expo and have a booth there selling our products. Team Zero's 2 star players comes with us to help promote, play a bit with the audience as part of the sponsorship contract, etc. We then get double the product sales we got before at this convention.

Over the next few months, Team Zero does really well in the EU LCS and makes the semifinals at worlds. We post about Team Zero on our social media platforms (and they post about us on theirs as part of the contract). We don't have as many likes and followers as the team, but over time we find our Facebook and twitter follower count increasing more than before. And many of them are also fans of the team.

By the end of the year, our sales team across Europe reports increased sales in several countries, especially the country where Team Zero's star player is from (let's say Poland, where sales have tripled). Due to our increased followers and fans on social media, we also find our non-eSports related promotion campaigns are more successful. For example, Team Zero's fans who didn't know about us before now join one of our launch parties held at a club with a famous DJ.

We calculate how much we spent on the sponsorship versus how much sales has increased. We cannot definitively link all of it to our sponsorship of Team Zero, but just by looking at the timing of sponsorship, countries where the sales increased and social media follower increase (where we can also prove our new fans are also fans of Team Zero)... we can REASONABLY CONCLUDE the sales increase profit is slightly higher than the sponsorship cost after a year. We also believe the increased support we have will last and may continue to grow. Therefore, we conclude the sponsorship is successful.

This is a simplistic example, but just imagine this happening on a larger and more intricate scale and it shouldn't be far off. Source: run a small marketing company and have helped clients negotiate sponsorships before (not eSports though, and smaller scale).