In a very energetic or forceful manner, for example, “Talking away to beat the band.” Also, to the greatest possible degree, for example, “The wind is blowing to beat the band.” Something being done thoroughly or furiously.
The use of “to beat” meaning “to surpass” is simply a modern use of “to beat” in its older military sense meaning “to defeat or vanquish.” The use of “beat” in other phrases equivalent in meaning (“to beat all,” etc.) dates back to the early19th century. In 1863, Charles Dickens wrote, “‘Well!’ I says, ‘if this don’t beat everything!’”
“Beat the band” first appeared in print in the late 1800s. At the same time, another “band” phrase, “when the band begins to play,” was in use, meaning “when things get serious,” or what might today be “crunch time.” It may be significant that both phrases arose at a time when recording technology was in its infancy and music was almost always heard live, whether in a music hall or at a concert in the park.
It has been suggested that the phrase originated from Banagher, an Irish town on the Shannon, where the novelist Anthony Trollope (1815-1882) spent some time as a Post Office surveyor. However, the town was chiefly famous for its inclusion in the saying, “That beats Banagher,” a common reaction to something extraordinary or absurd. Banagher was once a ‘pocket borough,’ meaning that the local lord employed most of the residents, who voted as he directed (they were ‘in his pocket’). It became a common joke in Parliament at the time to quip, “Well, that beats Banagher!” whenever someone mentioned a pocket borough where every resident was employed by the local lord.
However, it’s more likely that the origin of “beat the band” is simply musical. In the early 1900s, band concerts were popular and bands often played at ceremonial events. The band would be the most audible and therefore most conspicuous entity around. Any action or performance which outdid the band would have been remarkable.
But things are the same today. It’s darn near impossible to drown out an amplified rock band.