Could you please tell me how to understand the grammar of "based on" in this sentence? And what does it mean?

A wish originates based on what they think is good.

See a translation

It means that the wish was originated with the information/using their opinion of what they think is good. It comes from ‘basis’ like a foundation, you use something as the foundation to create a new version or thought.

‘Based on’ can have a meaning similar to ‘inspired by’ but usually uses more of the source. E.g.:
‘The movie was based on the book’ - the movie uses the same contents as the book but it’s a different version.
Or:
‘Based on what she said, I won’t go’ - using the information given from what she said, he decided not to go
‘I’m basing my opinion on this article’ I’m sourcing my opinion/using it as a basis/foundation