In economic terms, there can be two purposes for having these things. You can do them to try to increase the demand for your product, or you can do them to try to decrease what is called the "elasticity" of demand for your product.
Once thing that determines demand for a product is consumer tastes. If you can get someone popular to endorse your product, your product may become "cooler" and more people will want it.
Elasticity of demand is a measure of whether people will keep buying your product if its price goes up. You want demand to be inelastic because that means when you raise the prices, you make more money (people still buy your stuff). To make demand be inelastic, you have to convince people that your product is way better than other similar things. So you have someone endorse it to try to create that idea.
In business sponsorships and endorsements are two specific types of marketing communications used to promote a company and its products. Endorsement consist advertisement in which well known personalities express their appreciation and support for for the company and its products. Sponsorship consists of the companies providing financial support to events and activities such as TV sows, sporting events, and social service activities, in return for public recognition of the support provided by them. Sometimes the the people conducting the sponsored events also allow the sponsors the facilities for some direct advertisements.
Both these types of communication help to improve the customer support for the company and its products because of their association with the endorser or the event sponsored. The feelings people have about the people endorsing or the events sponsored are to some extent also transferred to the company and its products. The best ways to prove this will be to look at the experience of many companies who have increased the sales of their products by such means. Other direct way will be to conduct a market research to study how people exposed to such endorsements and sponsored events have been influenced.