logger (1) - a shell command interface to the syslog(3) system log module
For details see
-f, --file file
Log the contents of the specified file. This option cannot be combined with a command-line message.
This means that you try to read the message from
~/llog.txt. So it should be cleared "why" you can't find the text "any message" in this file.
If you read this carefully you would understand that all log messages are written to the syslog facility.
If you want to write the messages in one special file, you have to set up your syslog configuration to support this.
How to do depends on the syslog daemon you are using.
rsyslogd has very elaborated rules to control this kind of stuff.