This is a very good question. I don't have a definite irrefutable answer, but will mention some of the ideas usually thrown around when discussing this.

I'd like to start with a small correction. I would not say that Mendelssohn rediscovered Bach's music, I would say he helped reviving it. Bach's works weren't completely lost, they circulated in private copies and were some times played. I think these works mostly circulated among "music geeks," but they still were around.

Mozart knew of Bach, and admired his works. Christian Gottlob Neefe, one of Beethoven's teachers, was involved in the Simrock edition of the WTC. He had Beethoven studying the WTC, and Beethoven later recommended it as a great thing for musicians to study (both for technical skill in the keyboard and to see great compositional skills in action). Bach had very talented sons (they were pretty well known during their lives), and some knowledgeable students, who kept copies/originals of some of his works and to some extent kept his music alive. Gottfried van Swieten was a patron of Haydn, Mozart and Beethoven. He was a student of Kirnberger, who might have been a student of Bach (or maybe not, but in any case he was a big, big fan of Bach's music). Van Swieten was pretty knowledgeable about music, and Bach's was at least some times played/discussed because of him.

Bach's sons and students then had patrons, friends and their own students. Music geeks come in contact with other music geeks, and train new music geeks... Mendelssohn and others in his circle had came in contact with Bach's music in this way. Mendelssohn was not the first to want to try to organize a big performance of Bach music, but he was able to make it big.

The Mendelssohn's were doing well, Felix was a very talented musician and made a reputation for himself. He was about 20 when he conducted the St. Matthew Passion, but he was already a well read young man and had proven to be a superb musician (he composed THIS at 16, and one year before the super pianist Ignaz Moscheles considered he had nothing to teach young Felix). He managed to pull off the performances: getting the ball rolling and getting people to attend (family name, connections, and reputation), getting people to WRITE ABOUT IT, and he was also knowledgeable enough to make edited versions of Bach's work (the version he presented was abridged and re-orchestrated).

So, that happened. At the time, it was fashionable to know about old and obscure things (kind of a hipstery thing) and the German speaking world was trying to create some kind of unity. Musicology was being born, and the "canon" started to be created (see this previous answer).

A young hotshot from a well doing family presents the music of a great forgotten protestant German musician. Just the kind of thing the musical and the intellectual worlds wanted to hear. Top notch German music, and an opportunity to revive the former greatness. The ball started rolling, and it was a pretty good time to study music from the past.

Bach's music is very brainy, and the St. Matthew Passion is pretty big. Consider Mendelssohn's version was performed 5 years after the first performance of Beethoven's 9th, and 3 years after Beethoven's Große Fuge. Music of this caliber, composed how many decades ago? How come this stuff was not been played more often? That kind of music was very interesting, given the direction music was taking: big works, structure is super important. That is the music the great Beethoven liked, he was a very well respected composer, and in later years he was kind of worshiped (he became the ideal of what a [German] composer was meant to be).

You would probably be VERY interested in reading Celia Applegate's Bach in Berlin: nation and culture in Mendelssohn's revival of the St. Matthew Passion.

TL;DR

Young hotshot from a good family presents interesting music and manages to get people talking about it. It was not just about the music, but about the formation of a national identity: The Germans hated the French, and were interested in getting names for their collection of "greatest composers ever" to show just how great German anything was. Bach's brainy music fitted with the direction music was taking in the professional world.