First, a history lesson:
Peter Cornelius originally claimed that these "Three Bs" were Bach, Beethoven, and Berlioz. It was Hans von Bülow that then replaced Berlioz with Brahms, and Bülow did it with a little pun: since "B" in German means "B♭," he said that "My musical credo is in E♭ major, with three Bs in the key signature: Bach, Beethoven, and Brahms!" This automatically sets the stage that this statement isn't the most serious claim ever made.
Now, the actual answer:
Apart from the alliteration, the main reason it's these three composers is because they typify each of the three eras of tonal music.
Tonality was really codified in the Baroque era, which dates from (roughly) 1600 to 1750. Johann Sebastian Bach is a perfect figurehead for this era; in fact, most music historians choose 1750 as the end of the Baroque because that is the year of Bach's death.
After the Baroque came the Classical era, which lasts from around 1750 to approximately 1830. Beethoven, having died in 1827, is a solid figurehead for this era.
Then came the Romantic era, which lasts from around 1830 to around the turn of the 20th century. Brahms was chosen as the figurehead here. As for why Brahms should stand next to Beethoven, it might be important to note that von Bülow jokingly labeled Brahms's First Symphony "Beethoven's Tenth."
There are a few problems with this trio:
- Beethoven is really best understood as a transitional figure between the Classical and Romantic. In short, this trio under-represents the Classical era; we need a Bozart!
- Similarly, some (like Schoenberg) say that Brahms is better understood as a transitional figure between the Romantic and into the Modernist era. But this is less of a problem than it is for Beethoven, because Brahms was pretty firmly rooted in the Romantic.
- Notice that the list is entirely Austro-German. This is a problem in much of music history: since the original music historians were German, our views of music history emphasize the German tradition. Missing in this trio are the masters of the French, Italian, Russian, etc., etc. traditions.
- It privileges tonal music. There was something like tonality before the Baroque, but it was still being solidified; this list ignores everything before and including the Renaissance. Similarly, it ignores music after about 1900 (Brahms died in 1897).
As for whether it's helpful? I guess it's as helpful as anything would be that tries to summarize centuries of tonal music into three words :-)