At the time that the big announcement was made in the 90's, the scientific community immediately came up with a bunch of criticisms that weren't properly addressed in the original paper. They suggested terrestrial contamination, they suggested abiotic crystallization processes, and given the observed similarities with Earth fossils they questioned whether cross-pollination of life was possible. The scientists had no satisfactory answers and ALH84001 was dismissed by the press as a dud and the scientific community as a career-ending topic (like cold fusion is for physicists).

Still, in the years that followed further analysis was done by the original authors and their teams. Isotope ratios and captured atmosphere within and around the tracks confirmed that the fossil features were ancient and of Martian origin. Comparative characterization of terrestrial rock-worm fossils was done and it was noted that like the Earth fossils--and unlike abiotic processes--the ALH84001 features were of uniform size and were anisotropic on the sub-microscopic scale, consistent with fossilization of biological structures rather than geologic crystallization processes. Finally, testing showed that the inside of the ALH84001 meteorite itself never exceeded 40 degrees when it was blasted from the Martian surface, or when it entered Earth's atmosphere, so any biological life deep within the rocks could have survived (no one thought the fossils were live, but by implication Mars could have, and almost certainly was seeded by Earth, making it not surprising that we would see a the telltale signs of Earth microbes in a rock from Mars.

TL;DR: At the time of the press conference and the weeks after, a bunch of reasonable questions were raised about the ALH84001 work. In the years that followed a few of the scientists involved followed up with further tests which confirmed or at least did not disprove their martian-life interpretation of the fossils. But the rest of the world moved on.