Not much information to start on this one:

Check the flag05 home directory. You are looking for weak directory permissions
To do this level, log in as the level05 account with the password level05 . Files for this level can be found in /home/flag05.

The command ll (an alias for ls-alF) showed me that I had read access to ~flag05/.backup, and in there was a backup gzipped tar. Hopefully some goodies in here…

I unpacked the tar and found that it contained a folder called .ssh. This is used for secure shell authentication. That folder contained a private/public key pair and an authorized_keys file. The authorised_keys file is exactly the same as the public key file, so (assuming that the authorised_keys file had not been deleted since the backup) I should be able to ssh in using the private key, as long as it was not encrypted with a passphrase.

I copied the id_rsa file to ~/.ssh and tried to connect using:

ssh flag05@localhost

Bingo!