Anguirus' skeleton was secretly included in Godzilla: King of the Monsters, and the movie's credits may have revealed what killed him. In classic Toho movies, Anguirus is one of Godzilla's most recognizable and iconic allies. Anguirus fought at Godzilla's side as he went up against King Ghidorah and Gigan in Godzilla vs. Gigan, and was one of several kaiju to join him in the battle with King Ghidorah in Destroy All Monsters.
In Godzilla: King of the Monsters, Monarch heads down into the Hollow Earth tunnels to revive Godzilla, where they come across an ancient underwater civilization. Upon further study, they discover that this place is actually older than both the Romans and the ancient Egyptians. Paintings on temple walls indicate that the people of this long-forgotten city once worshiped Godzilla as an actual god. Apparently, Godzilla wasn't the only Titan who lived there, as a Titan skeleton can be spotted in a single shot. The special features on the home video release provide a better look and all but confirm that the skeleton belongs to Anguirus.
It's never said what happened to Anguirus, but the answer could be found in the credits of Godzilla: King the Monsters. If viewed carefully, audiences may notice that the redacted text in the credits tell a long and detailed story about the history behind the relationship between humans and Titans. It's said that Titans and humans went to war after humans tried to enslave Titans. Though Titans are obviously superior to humans, it is possible that some Titans were killable. This could suggest that the human-Titan war that ravaged the Earth was the cause of Anguirus' death.
Even if Anguirus wasn't a victim of the war, other events followed which could have caused Anguirus to meet an early end. The war triggered an unidentified "massive cataclysm" which destroyed the human civilization. If it was powerful enough to wipe out an entire civilizations, it's not implausible that some Titans may have been casualties as well. A consequence of the "cataclysm" is that it resulted in an Ice Age, along with other geological catastrophes. The credits specifically state that Titans weren't invulnerable to these disasters, which meant that some could have been killed. The rest of the Titans went into hibernation, but others, like Anguirus, may not have been so lucky.
Any of these events could be the explanation for Anguirus' fate in the MonsterVerse. Unfortunately, it's unlikely that fans will hear any confirmation anytime soon on what happened to him, especially since his name hasn't even been officially confirmed, on account of rights issues. For King of the Monsters, Legendary only had access to Godzilla, Rodan, Mothra, and King Ghidorah. However, this could always change for another movie, like Godzilla vs. Kong, if Legendary were to acquire the rights. This would allow the MonsterVerse a chance to further explore the ancient history of the Titans and maybe even another member of Anguirus' species.