Similarly to how Germany attempted to cripple the R.A.F., and how she attempted to destroy British morale and industry, via the Luftwaffe, Germany attempted to starve Britain out by sinking incoming supplies with U-boats.
"The only thing that ever really frightened me during the war was the U-boat peril." - Winston Churchill.
In the beginning of the war, the U-boats did their damage. Between September 1939 and June 1940, 702 merchant ships were sunk by the U-boats. But, fortunately for Britain, they were unable to get closer than a few hundred miles from the Coast of Britain.
Then, France and Norway fell to Germany, and this all changed.
More U-boat bases were brought into the picture, and those in France greatly reduced the U-boat's commute time, allowing for more of them in the water at once, as well as increasing their range.
U-boat production rose and rose, as did their effectiveness. Over the course of the war, Britain lost 60 percent of its normal peacetime cargo.
The Germans calculated that in order to starve Britain out, they would have to destroy 750,000 tons of incoming cargo to Britain. In September of 1939, Germany sunk only 200,000 thousands tons of said cargo. But this number rose.
In April, 1941, 700,000 tons of cargo were lost -- Germany was nearing her goal.
However, thanks to better usage and protection of the convoy system, these sinkings began to decline. Another factor was that the German naval code was now being read, thanks to British crypt analysts.
While the Battle of the Atlantic was by no means over, the destruction of Britain was averted.
Sources: The Historical Atlas of WWII | Alexander Swanston and Malcolm Swanston, World War II The Definitive Visual History | DK