What does “Domain transfer initiated and awaiting approval” transfer status mean?

It means that there are not known issues at this time, transfer is processing as normal.

According to ICANN's 'Policy on Transfer of Registrations between Registrars', losing registrar has 5 days to release the domain, from the moment they get transfer out notification. You might want to contact them and specifically instruct to release the domain to us, however in most cases this is a highly automated process, so there are no special actions required at this stage of transfer.