Here the individual is given funding by that organisation, and in exchange they wear the company logo on their . This then means that the thousands of people watching that sporting event will all see the logo and awareness will increase for the business. Likewise companies can sponsor the teams themselves in which case they might name the stadium after that company, or they might use certain branding in the stadiums.
If you wish to sponsor a company then there are many different avenues for you to go down, but if you want to use a smaller event or organisation then they will likely require you to provide the logo t-shirts yourself. This way you can pay them to wear the that you have made using t-shirt printing and you can tightly control the message they put out about your company. This is a great use of custom t-shirts as it means your logo will be seen by a maximum number of people with relatively little work on your part.