The main reason that the British sent reinforcements to America at this time is simply that they had become available. For the British, the War of 1812 had taken secondary importance to their fight against the forces of Napolean. However, Napolean's defeat at the Battle of Paris in March 1814 and his subsequent exile meant that the British could refocus their attention on a new attempt to bring the war in North America to a quick end. As a result, shortly before the Battle of Bladensburg, huge numbers of battle-tested British soldiers were transported from Europe to America.
This was a great threat to the American forces. While the British had been focused on their wars in Europe, the American military was able to more or less hold its own against the British forces in North America. However, this new threat of numerous and well-trained reinforcements spelled a potential disaster.
Most of the new British reinforcements arrived under the command of Major General Robert Ross. Ross saw the opportunity of forcing an American surrender by attacking the political capital of Washington DC. Ultimately, too few American forces were in place to counter the advancing British and the Battle of Bladensburg was more of a rout that left Washington open to being sacked. While this did not have the effect of bringing the conflict to a decisive end, it was still a huge defeat and embarrassment to America.