How to restore symmetric key on the new database?

There is no built-in solution in SQL Server. A symmetric key that was created without specifying the source and identity can never be scripted or copied.

That means, in your case you have to decrypt and re-encrypt on the fly while copying the data.

You could also create a new key in the current database (specifying source and identity) and decrypt and new-key-re-encrypt locally. Then you can move the encrypted data and recreate the key in the new location, as you have all the important seed data available for that new key.