Alright, most of his backstory is rooted in Undertale so just fair warning.

In a parallel universe to Undertale there was an accident at the CORE. Not only was Doctor Gaster affected but many other scientists on site as well. 2 of them were Cosmo's parents, Sagittarius and Summer. But, Summer was a human. You see, in this parallel world the monsters, while losing the war, they didn't spend as long stuck in the mountain. They were able to break the seal much sooner and restore peace among the two races sooner. So, when the accident happened Humans and Monsters have been working together for quite some time. Some even fell in love. Sagittarius and Summer were one of those couples. Usually, the biological differences between monsters and humans would make it impossible for them to have kids together, but as them being scientists they were able to discover a solution. However, it would only work for them, but it was a step in the right direction. The end result was a miracle, Cosmo. Or, at the time known as Leo.

At the time of the accident Sagittarius and Summer were showing to their co-workers their child while at the lab. Even as the alarms were sounding that something was horribly wrong neither of the parents were able to get their son out of the way. Everyone at the lab was completely wiped out and spread across time and space, and destroying their universe in the process. Gaster, in the void. And Cosmo in an alternate universe. The fate of the other scientists are unknown, but are assumed to have been scattered as well.

As Cosmo grew up in the alternate universe people were not so kind to him. It was obvious to the other kids at the orphanage he ended up in that he was not completely human. His hair was a galaxy after all, infact, that's where his name came from, Cosmo. For being different, he was bullied. Cosmo, where he was flung was an alternate universe parallel to his own. But, he had been flung foward in time, when this universes monsters had recently broken the barrier.

Time passed. Cosmo grew closed off to the world. Now, here is the part of his story where I grow uncertain what exactly happened so, we're going to skip it for the time being. The important thing to note is that Cosmo gained the knowledge of what happened to his universe, and the very fact that he's not from his universe. He also found and harnessed his powers.

Now, he travels space and time in hopes of finding his lost family. Assuming what happened to him was not a special case. He will find friends along the way and have great adventures. Right now, it's been 6 years since he's started.