The AGPL confuses a lot of people on a lot of projects. It's actually really simple: If you modify the package (RethinkDB itself, in this case), you have to release your changes. That's it.
That's not precise. If you modify the package and distribute it to your users, they must have access to the source code (including the changes, which also become licensed under the same terms). Up to this point, it's the same as with GPL.
AGPL also enforces code and changes to be distributed to the users if they directly consume your database over the network. This would be the case, for example, of a database-as-a-service.
Put it simply: you use an AGPL database, internally, even for your SaaS: you're fine. You either modify RethinkDB and ship it as a product or you provide a RethinkDB or RethinkDB-derived database-as-a-service, then you have to also provide source code to your users.
