They keep everything in balance.
Sharks keep populations of other marine life healthier by picking off the sick and dying individuals. Where there are more sharks there are more fish. Over 1 billion people rely on seafood as their primary source or protein, many of them living in underdeveloped countries or islands.
Sharks are a keystone species. They feed at the top of the oceans' fragile and intertwined food chains. When they are removed the entire thing collapses.
They keep the best grazing areas from becoming overeaten.
The intimidating presence of tiger sharks can cause important spatial distribution among other species in the area. For example, in Australia, dugongs and sea turtles will choose to feed over a broader area to the sharks, rather than spend all their time grazing on the area containing the best sea grass until it becomes barren.
And even help coral reefs!
Sharks aid coral reefs by keeping their fish populations properly balanced. This in turn allows the small predatory fish to not wipe out the herbivorous fish that eat algae off of young corals.
But how do these things affect us?
In more ways than you might think!
Let's look at an example of the effects lower shark populations have had on certain habitats.
On the east coast of the US sharks have been wiped out so heavily (50-75% decline in the last 15 years for all species, excluding mako) that they've been labeled "functionally eliminated," unable to perform their natural role in the the ecosystem. The populations of their natural prey, rays and skates, have soared. These rays have decimated the local scallops, clams, and oysters so efficiently that multiple long-standing fisheries have collapsed.
BUT these scallops, clams, and oysters (also known as bivavles) perform their own important natural role in the ecosystem too! They feed on phytoplankton, improving the quality of the water, and act as a natural filtration system. In their absence, areas can have water with such low levels of oxygen that they become dead zones.
No matter where we live in the world, half of our oxygen comes from the sea. And considering we're eliminating our other primary sources of oxygen (forests and trees) pretty quickly we should be VERY concerned about the effect a shark-less ocean.
There is no Planet B.