When sport/event organizations seek sponsors who will exploit the sponsorship, what benefit are they seeking to gain?

they want to generate more income from doing so, the greater the allowance the more income they will generate for the event they intend holding.

Explanation:

Exploiting of sponsorship would be described as allowing a sponsor to utilize every opportunity in the event to promote their brand.

When event organizers look out for sponsors who would handle their event and allow them explore it, it'll mean they want to generate more income from doing so, the greater the allowance the more income they will generate for the event they intend holding. For them to allow a particular brand to exploit the event would mean that brand would virtually take care of all expenses for the event since it'll be the face the audience would see when they come to the event. It is a way of generating more income for events.