"Big Ancestor" by F. L. Wallace. You can read it at the Internet Archive. The anthology you read it in could be Galactic Empires edited by Brian Aldiss.
• Humans were traveling to other planets trying to decipher their big destiny.
"Plus or minus a hundred thousand years, we can still get something that might be the path of a spaceship attempting to cover a representative section of territory," said Kelburn. "However, we have other ways of dating it. On some worlds on which there are no other mammals, we're able to place the first human fossils chronologically. The evidence is sometimes contradictory, but we believe we've got the time right."
Taphetta waved a ribbon at the chart. "And you think that where the two ends of the curve cross is your original home?"
"We think so," said Kelburn. "We've narrowed it down to several cubic light years—then. Now it's far more. And, of course, if it were a fast-moving star, it might be completely out of the field of our exploration. But we're certain we've got a good chance of finding it this trip."
• On each planet where humans had arisen there were monolithic ruins of a previous, giant race.
Emmer smiled, unsheathing great teeth. "You've never seen any pictures? Impressive, but just a camp, monolithic one-story structures, and we'd give something to know what they're made of. Presumably my world was one of the first they stopped at. They weren't used to roughing it, so they built more elaborately than they did later on. One-story structures and that's how we can guess at their size. The doorways were forty feet high."
• While traveling, they were having trouble on the ship with rodents that were evolving rapidly because of exposure to radiation
"It was probably brought in with the supplies," said the biologist. "Considering how far we have come, it may have been any one of a half a dozen planets. Anyway, it hid, and since most of the places it had access to were near the outer hull, it got an extra dose of hard radiation, or it may have nested near the atomic engines; both are possibilities. Either way, it mutated, became a different animal. It's developed a tolerance for the poisons we spray on plants. Other things it detects and avoids, even electronic traps."
and they were starting to use weapons.
The trap setting was changed and several animals were taken. Physically, they were very much as Halden had described them to Taphetta, small four-legged creatures with fleshy antennae. Dissection revealed a fairly large brain capacity, while behavior tests indicated an intelligence somewhat below what he had assumed. Still, it was more than he wanted a pest to have, especially since it also had hands.
The biological mechanism of the hands was simple. It walked on the back of the front paws, on the fingers of which were fleshy pads. When it sat upright, as it often did, the flexibility of the wrists permitted the forepaws to be used as hands. Clumsy, but because it had a thumb, it could handle such tools as a knife.
He had made an error there. He had guessed the intelligence, but he hadn't known it could use the weapon he had put within reach. A tiny thing with an inch-long knife was not much more dangerous than the animal alone, but he didn't like the idea of it loose on the ship.
• Eventually, the kicker is deciphering the words on the ruins
"The translation is complete," announced the machine.
"Go ahead," Meredith ordered.
and learning that humans descended from rodent-like things that evolved rapidly on the spaceship of this prior giant race.
"We didn't guess that next to the hull in outer space and consequently exposed to hard radiation," the message went on, "those tiny creatures would mutate dangerously and escape to populate the planets we landed on. They had always been loathsome little beasts that walked instead of rolling or creeping, but now they became even more vicious, spawning explosively and fighting with the same incessant violence. They had always harbored diseases which spread to us, but now they've become hothouses for still smaller parasites that also are able to infect us. Finally, we are now allergic to them, and when they are within miles of us, it is agony to roll or creep."
