Hungarian, IMO, is a “purpose” oriented language being very precise in expressing what someone will due to what end (we even have an “objective” /tárgyi/ verb declination to express that someone is not doing something in general but in particular with a given object).

So: you wash down the mirror because you take the dirt down from its surface;
you wash up the floor because you take the dirt up from the floor (to the bucket);
you wash away the dishes because you take away the dirt from them (to the sink)
and of course the “meg”: it roughly means to do something to the end or to the point it’s considered finished or until the expected or supposed outcome is achieved.
e.g: “tanulni” – to study / “megtanulni” – to apprehend completely;
“lovagolni” – to ride a horse (or something eg: a wawe or trend) /
“meglovagolni” – to ride the horse until it’s “owned” and you have full control of it (or ride the wave until it hits the shore or ride the trend until you have the results you expected: profit, sucess etc.
“szeretni” – to love / “megszeretni” to be over the process that produced love or liking

Hope that shows the logic
Strange as it seems :)