That script - .bash_logout - is executed by bash when login shell exits, and that code from your question intends to clear the screen to increase privacy when leaving the console.
SHLVL is a environment variable which came from "SHell LeVeL" and lets you track how many subshells deep your current shell is. In your top-level shell, the value of $SHLVL is 1. In the first subshell, it's 2; in a sub-subshell, it's 3; and so on. So SHLVL indicates how many shells deep the user is. If the level is 2, you must type exit, then logout to exit.
So, if "$SHLVL" = 1 i.e. if you are in the top-level shell, then...
[ -x /usr/bin/clear_console ] is another test and means something like: test if the file /usr/bin/clear_console exists and is executable.
Because of && this command: /usr/bin/clear_console -q is executed only if [ -x /usr/bin/clear_console ] succeeds with success.
And finally: what does /usr/bin/clear_console mean? From man clear_console:
clear_console clears your console if this is possible. It looks in the
environment for the terminal type and then in the terminfo database to
figure out how to clear the screen. To clear the buffer, it then
changes the foreground virtual terminal to another terminal and then
back to the original terminal.
clear_console is very approached to clear command which can be used in any terminal/console.