In bash (and most other shells) you assign variables without the $, so PATH=something assigns the variable PATH to the string "something".
In Unix/ Linux, PATH is a string variable that contains a list of folders separated by colons (:). For example on my laptop this is my PATH:
> echo $PATH
/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin
This means that any executable in those folders is accessible anywhere without having to type the full path. You will typically add things to the PATH in your .bashrc file, so they get added when your shell starts up. If you use the which command you can see where a command lives, it will be in one of these folders:
> which rm
/bin/rm
To add a new folder to the PATH, you re-assign PATH to be a string of the new folder, followed by a colon and the previous value of PATH.
In you example, you are adding $HOME/.rbenv/bin to the start of PATH. $HOME will be expanded to your home directory.
To understand this better we can do something like this:
> echo $PATH
/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin
> export PATH="$HOME/.rbenv/bin:$PATH"
> echo $PATH
/Users/javanut13/.rbenv/bin:/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin
As you can see the rbenv folder was added to the front of the PATH variable.
export puts the variable into the environment of the current shell, which means that other commands started in that shell (subprocesses) also get this change. For example if you don't do export and then run a bash script that uses ruby, it will fail because it doesn't have the ~/.rbenv/bin folder in its path.