Red XIII only has one eye in Final Fantasy 7 and Final Fantasy 7 Remake, and neither game gives an explanation as to how he lost the other. The story of his injury can only be found in FF7's supplemental materials, though it could appear in future entries of the FF7 Remake series.
In the original FF7, the members of Avalanche accidentally free Red XIII while trying to save Aerith from imprisonment in the Shinra Headquarters. He joins up with the party, as he sees strength in their numbers. Once the party leaves Midgar, he sticks with them throughout the rest of the game in order to protect the world from Shinra, and players learn more of his backstory while in his home of Cosmo Canyon. Though he plays a relatively minor role in the overall story, he gets quite an important moment all to himself: The last canonical scene in the original FF7 timeline involves Red XIII traveling to the ruins of Midgar with his cubs, though this event was changed in the remake, as FF7 Remake's party created a new timeline.
When Cloud meets the dog-like character for the first time in both FF7 and FF7 Remake, Red XIII is missing his right eye. The games never explain his wound, but the answer can be found in the Japan-exclusive Final Fantasy VII 10th Anniversary Ultimania book, as translated by The Lifestream.
Red XIII's Eye Loss Is Only Partially Explained Outside Of FF7
In the Before Crisis video game - a Japan-exclusive prequel to FF7 that will be soon be localized as part of the Ever Crisis mobile game - the Turks capture Red XIII as he is participating in a ritual with Deneh, the last remaining female of his species. Red XIII sacrifices himself to their capture so that Deneh can escape. Both of his eyes are intact during this prequel chapter, which takes place shortly before FF7 proper begins.
Final Fantasy VII 10th Anniversary Ultimania explains Red XIII lost his eye while being held captive by Hojo, whom the Turks brought him to. In FF7, Red XIII talks about having a score to settle with Hojo, which likely refers to the eye he lost. It's unclear whether the eye was purposely removed as part of an experiment, whether it was injured by accident, or whether it happened during an escape attempt.
Red XIII was reduced to NPC party member status in FF7 Remake, as he was introduced too late in the story to be playable. It's likely Red XIII's story will be expanded in future entries in the FF7 Remake series, however, and fans might finally learn the tale of how exactly he lost his eye, as told by the good boy himself.
Source: Final Fantasy VII 10th Anniversary Ultimania via The Lifestream