Long ago and far away, in an age long since past, MATLAB did not have function handles. There were no anonymous functions. There were no nested functions.
So the writers of MATLAB came up with a way to pass in arguments through functions like ODE45, like FMINSEARCH and FMINCON, etc. This was because all of these functions sometimes need to allow extra arguments to be passed in, essentially through those tools.
The idea was that if your function has "extra" arguments beyond those expected, then they are added as extra arguments to the optimizer or the ODE solver. Then the solver is smart enough to attach them to the function call itself when that call is done internally. It worked, but it was a programming kludge, and a massive one. It was confusing to users. To be honest, global variables were a better solution, and I hate global variables.
So at some point, the powers at MATLAB realized that this was indeed a bad idea. So, for years they warned us users that the option would be "removed" from those tools, but they had by then provided us with function handles, with nested functions. They had provided us with better ways to solve the problem.
At some point, they removed the option from the documentation for those functions, although I think it still worked for some years, so that legacy code would not fail. Without testing some code, a quick scan through the code for fminsearch suggests that it still (in R2019a) allows that legacy form of call for extra arguments, although I saw nothing in the help that said it was legal. I also just scanned the code for ODE45, and it too looks like it is allowed as a hidden option. Without testing, I would not state this to be true though.
You must remember that it is important that legacy code should generally work, if at all possible. I have code that is now over 30 years old, code that still works nicely. Unfortunately, that means sometimes people who never saw this capability will not know it ever existed. And some people like you will be confused when looking at legacy code, perhaps written by a MATLAB neanderthal like me.