In order to determine the importance of nature's top predator in an ecosystem, you need to study the entire ecosystem which means you can't just look at what tiger sharks eat (dolphins, fish, turtles) but you also need to look at what the tiger shark prey consumes (smaller fish and sea grasses).
The area being studied in the video is in a remote location of Australia and has had little impact from humans. As the moderator states, the area appears similar to what one would have found hundreds or even thousands of years ago. It has large populations of sharks, fish and undisturbed sea grass beds.
Through the use of clever research techniques, researches learned that the mere presence of sharks in an area determined the overall robustness of area sea grasses. In areas where tiger sharks were not present, sea turtles would move in and graze on the grasses causing it to be thin and not very dense. But in areas where tiger's were present, the turtles would avoid the area thus allowing the sea grass to grow long and lush.
The conclusion was that if we were to remove tiger sharks from an ecosystem, the ecosystem would change and as noted in the video, the change would "definitely be for the worse." The mere presence of tiger sharks ensure that the grasses have a chance to grow into dense beds thus providing shelter for smaller fish and crabs. These smaller fish and crustaceans eventually grow up and provide food for larger animals, including people.
This a wonderful short video that succinctly explains the importance of tiger sharks in our waters. It was posted by the National Science Foundation and is 3 1/2 minutes in length.