Ta-dah! comes from the Bulgarian or Slavic words for "ta + da" (та да!)meaning "that there". It is an exclamation used in magic shows (prestidigitation) by magicians to announce the conclusion of the trick or the illusion to the audience.
The equivalent to 'Voila!' in French. It was likely a Bulgarian or Russian magician (definitely Eastern European) traveling in the United States that said it (likely in the late 1800's when Eastern European immigrants started flooding to American shores).
An American likely heard it and thought it sounded usuable for showmanship, without knowing what it was. та да! Then magicians everywhere started using it as a handle or a gimic because it sounded more impressive than saying "There you have it!" in English.
The Bible in Bulgarian, shows this sort of usage: "Behold!" "Voila!" and "та да!" Mean roughly the same thing. It is often used as an introductory to a sentence, mainly where God is speaking, but by itself is showmanship flourish.
The Bible in Bulgarian is written in Old Church Slavonic. Since Russian & Slavic culture manifest traveling circuses & magic shows in abundance, they a clever saying for presentation purposes. Once it reached America, however, the nuances of the actual original meaning in that culture changed for American ears to promote magic with magic sounding words. So, thanks to an ingenious Eastern European magician, magic got what it needed to become a permanent part of American pop culture.
(Magician pulls a rabbit out of his hat and exclaims in an upbeat, loud voice) Tada!