This word is just too much, its meanings are:
finding, switchboard, operator, real-estate agent, arranging etc.
Ok, let's look at some examples:
Sie haben sich durch die Vermittlung einer Agentur kennengelernt.
They met through an agency.
Ich habe das Zimmer durch die Vermittlung eines Freundes bekommen.
I got the room through a friend.
Hey, it's actually pretty simple, whenever I get something via assistance/help from someone else I just use durch+die Vermittlung+my helper in Genitiv. But:
Geld,das man für die Vermittlung einer Wohnung bezahlt.
It probably means money you pay to real-estate agency for finding you an appartment. (I'm not sure though)
Hey, it's actually pretty simple für+die Vermittlung+service done in Genitiv is actually a service someone did for me.But:
Vermittlung, geben Sie mir Göhren 487
Operator, give me Göhren 487 please
So, what can I conclude from all the examples?
Vermittlung is in fact 'an operator', that assist you in some way.Let's downgrade examples above:
They've come to know each other with the help of agency operator.
My friend-operator helped me with apartment
Money you pay to apartment operator (the one that finds apartments).
I know it sounds atrocious and oversimplified but can I get away with it?