The typesetter for the 1911 edition accidentally replaced that with to, undoubtedly because of the fairly high expectation in English of the phrase “testify to”. (It occurs three times in the Book of Mormon text: once in 2 Nephi 27:12 and twice in Alma 18:1–2.) But this 1911 mistake was not continued in subsequent LDS editions, simply because it resulted in a stranded predicate later on in the passage (“manifested plainly of the coming of a Messiah”).
Summary: The 1911 change from that to to is simply a typo and was due to the commonness of the phrase “testify to”.