Dream Devourer is the result of Schala entering the time warp in the Ocean Palace that relocated herself, Janus, and the three Gurus: Belthasar, Gaspar, and Melchior. While Janus ended up in the middle ages, Melchior ended up in the present, Belthasar ended up in the far flung future, and Gaspar ended up at the end of time, it was unknown where Schala ended up.
In the DS version of Chrono Trigger a bonus series of dungeons exist that result in discovering Schala's fate. After completing the time distortion dungeons Crono and crew travel to a new location, beyond the boundaries of time and space. There they meet a version of Magus from an alternate universe where Lavos has already been defeated. This is likely a Magus that has come from a full ending of Chrono Trigger where Lavos is defeated and Magus leaves in search of his sister, Schala. It is here, beyond time, that you discover what became of Schala.
Schala ended up beyond time where she merged with a defeated Lavos. This merger was the result of Schala's desire for the end of all things. Now she has become one with Lavos to create a new entity, the Dream Devourer. Visually this merging of the two entities is represented by Schala floating in a red orb located ontop of Lavos.
The other Magus attempts to free Schala only to be defeated in one attack by the Dream Devourer. It is after this that the boss fight starts.
After the fight ends, with Dream Devourer defeating the party with a single final attack, Schala reveals that she cannot be defeated, nor can she be freed from her current state, no matter how powerful Crono and his teammates are or ever become. It is not thoroughly explained what exactly has happened to cause the incredible increase in power of these two entities after the merging, but it is clear that the Dream Devourer intends to end all of existence, may very well have the power to do so, and cannot be stopped by any normal means.
Schala, who currently retains some amount of individuality from the Dream Devourer still, tells the party that it is best if they leave here and forget that any of this has ever happens. Afterwords she returns the party to the present time; though it is unclear what, exactly, happens to the party members after this.
The Magus from the other universe is also removed from Dream Devourer's presence. Magus, however, ends up in an unfamiliar location and has lost all memory of who he is or where he is from. It is unknown what happens to him after this point. The Dream Devourer however continues to exist beyond this encounter and goes on to become the final boss in the alternate universe sequel, Chrono Cross, and adopts a new name, Time Devourer.