In the Angel Beats! universe
The anime didn't tell us much about what happened to the people who passed on, except for the ambiguous scene at the end. There is a chance that the ongoing visual novel will explore this topic a bit. For now, we can only refer to side materials, such as the third drama CD released with the sixth limited edition BD/DVD volume. In the drama CD, everyone reincarnated as a human being with a similar look (except one guy), but not with a similar personality (except a few). They all met in the Afterlife World for the second time after dying again, and they were able to recognize each other and remember their previous lives.
In the Keyverse
SPOILERS ahead for the following series: charlotte, clannad, little-busters and one.
There are recurring themes and concepts that connect together the worlds of Angel Beats!, Charlotte, Clannad, Little Busters! and One. Note that One is generally considered to be part of the Keyverse because
[m]uch of the staff that created the game later became the founding members of the visual novel brand Key.
In Clannad, there is the Illusionary World:
This world is later revealed to have been created by Ushio, who died regretting having left her father alone. She therefore became the girl in this world with the hopes of meeting her father again and collecting enough light orbs to make a miracle possible. [...] In theory, the Illusionary World is the after-life; when a person dies they are still connected to the real world, yet they are free to create their own world.
In Little Busters!, there is the Artificial World:
When the artificial world was created, her wandering soul which still had a lot of regrets and unfulfilled wishes entered the artificial world. [...] [Riki] got involved with Saya so much that her existence caused him to stray away from the other Little Busters members. At some point, Kyousuke realized that he has to make Saya leave as soon as possible in order to come back to the original purpose for the creation of the world[: to make them strong enough to deal with what will happen when they awaken in the real world].
In One, there is the Eternal World:
The Eternal World is a place similar to the afterlife where a person's "other self" awaits. Anyone can access it, though it is only accessible once one has lost his or her grounding in the real world. It may be necessary to form a pledge with someone in the real world as a guide to the Eternal World, but one's memories surrounding the pledge become vague. A grace period can be granted between when the pledge is made and when someone goes to the Eternal World. Once the process has started, nothing can prevent someone from going to the Eternal World, and returning to the real world is difficult. Someone about to leave to the Eternal World starts to be forgotten approximately one week before going, and the amount of time before someone forgets differs based on how much someone thinks about the person leaving. However, he or she is remembered the moment that person returns. If a strong emotional bond is established in the real world before leaving, a person in the Eternal World can be returned after approximately one year.
In Charlotte, there is The World that is No Longer Here, which is a world where people with superhuman abilities exist, and regrets ensue. Otosaka forms a pledge with Tomori to loot the abilities of everyone in the world in exchange for her love, but eventually loses memories of the world with ability users.
Then recall that the main setting of Angel Beats! is the Afterlife World:
Despite the numerous enigmas that surround the world, it serves as a sort of second life for teenagers whose lives were filled with despair and pain. They are brought there after they have died and, alongside others, try to live there with fulfilled existences to make up for their pain. After, they disappear as their happiness has been granted. [...] Or, it could be simply said that time does not exist in the Afterlife and teenagers who died with regrets can go there, regardless what time of death or time period said person is.
In summary, all these worlds are for people who have regrets and lost memories surrounding the regrets.
So what happens in the Keyverse when a person "passed on" from one of these worlds? They return to the real world, and in the case of Tomoya in Clannad, he returned to a past point in time after collecting enough light orbs to grant his wish.
Based on the recurring themes and concepts in the Keyverse, this is the sequence of events that unfold after Kanade and Otonashi passed on from the Afterlife World:
- They had established a strong emotional bond before leaving to the Afterlife World, which ensured their eventual return to the real world. [One]
- They had been helping people achieve happiness by fulfilling their goals, so the light orbs that appear in the OP and in episode 13 granted their wishes and brought them back to a time before the point of no return so they could meet each other and be together. [Clannad]
- Since coming to the Afterlife World, Otonashi and Kanade had grown stronger and opened up their heart more, so they would be in better shape when they return to the real world to overcome their tragedies (abdominal injury and heart failure) and change the outcome of the adversities favorably for them. [Little Busters!]
- They were able to recognize each other thanks to a song of a certain musician, which they somehow retain the memories of. [Charlotte]
Interestingly, Otonashi Yuzuru (音無 結弦) means "no sound" and "fastened strings", Hatsune (初音), the name of Otonashi's little sister, means "first sound", like the first sound of heart beat of a person long thought to be dead, and Kanade (かなで/奏) means "to play an instrument". Playing the Meaningful Name trope, Kanade "plays" out the tunes of her romantic encounter with an Otonashi whose love string is fastened.