How do I transfer iCloud files from old iPhone to my new iPhone?

Did you restore the iPhone from a backup of the old one, or did you use Quick Start? Your voice recordings should have come over as part of the backup if you were using iCloud. Do you have Voice Memos turned on in iCloud?

Are you referring to iCloud Drive? If so, you log into the same iCloud ID on the new phone and turn the particular settings on that you want, such as iCloud Drive, Contacts, Calendar, etc.

Sign into iCloud on the new phone and toggle on the services you wish to sync:

Settings > Apple ID > iCloud