If you set a breakpoint in a function the debugger will stop there and you can inspect variables. You can also step in a variety of ways (step, step in, continue execution, etc).
The example provided in the blog post just seems to be encoding your own custom breakpoint mechanism that is available at runtime. It's not uncommon to use such constructs in a language to enable higher logging levels based on either a file (as in the example) or the setting of an environment variable. You can certainly do this in MATLAB jsut as you would any other programming language that can read the filesystem or check the value of environment variables. What you do when you hit that breakpoint is up to you -- continue execution, log a diagnostic message, output something to the screen, etc.