It is true! This is part of Table X of the Twelve Tables. The Twelve Tables were the oldest Roman laws, traditionally dated to 450 BC. The reason for it is partly practical (the city wasn’t very big and didn’t have a lot of room for burials) and religious/psychological, based on how the Romans saw their city as a religious space.

From a religious point of view, it wasn’t really the walls themselves that were the problem. The issue was that anything within the boundaries of Rome (the “pomerium”) was consecrated land. This goes all the way back to the legend of Romulus and Remus - Romulus ploughed a line demarcating the boundaries of the city, but Remus jumped over the line and polluted the consecrated space, so Romulus killed him. Historical Romans felt the same way. Inside the pomerium (and, more literally, inside the walls that were built over it) was a religious space, so certain things couldn’t be done there, including burials.

The Twelve Tables are pretty vague about this (…and everything else, really). The third provision of Table X says, simply, “No burial or cremation of a corpse shall take place in a city.”

Some Roman authors tried to elaborate on this. Cicero guessed that bodies couldn’t be cremated in the city because of the risk of burning the whole city down (On the Laws/De Legibus, Book 2.23), although he noted that lots of famous people were sort of illegally buried within the walls just because they were rich and/or famous. He doesn’t say why regular people had to be buried outside the walls, but it was probably too obvious to mention (the religious reasons that everyone knew and he didn’t think he needed to explain!).

Gaius (a legal scholar from the 2nd century) explains that if you bury a body on your own property, it becomes a religious site (Institutes, commentary 2.6). That’s different from burying someone within the city walls, but it’s important for this question because a burial site is a religious site. Relatedly, Paulus (maybe 4th century) explains “It is not lawful for a corpse to be brought into a city, lest the sacred places of the latter may be polluted” (Opinions, Title 1.21) So the issue is both that a burial site is now a religious site and cannot be disturbed, which would be a problem for the spaces within a city; but also that a consecrated site would be polluted in a religious sense if a body was buried there.

Practical concerns important too. Maybe the original idea of “pollution” was simply that they didn’t want dead bodies stinking up the city and causing diseases. The practical reasons then became codified in religion and law as the city developed. “Rome” itself, in a legal and religious sense, was rather small, so dedicating some of the space inside the city walls for burials would take up valuable space.

Burial/cremation sites could be a few miles outside the walls, or even right up against the walls (technically legal, the best kind of legal). There were lots of burial monuments along the Appian Way, and eventually people started using the catacombs to bury the dead. The same was true for other Roman cities, and cities elsewhere in the ancient world - long before Rome, in Jerusalem and Athens the burial grounds were outside the city walls as well.

For more info about pollution, see Pollution and Religion in Ancient Rome by Jack J. Lennon (Cambridge University Press, 2014). It’s not specifically about burials, but it does dive into the issue of the practical and religious meanings of “pollution”.

There has been a ton written about Roman religion in general, but a good place to start is Religions of Rome by Mary Beard, John North and Simon Price (Cambridge University Press, 1998).