The general scenario is that users upload documents to a document list and adds keywords to that document. These keywords are stored in a keyword set (if I understand things correct). Users might spell keywords wrong, and I as an administrator like to correct these keywords. Also there might be synonyms which could be merged to a single keyword, all in the purpose of making search better. But where do I find all of these keywords or the keyword set?
Very easy: The managed metadata service.
You can just hop to your Central Administration > Application Management > Manage Service Applications > Managed Metadata Service.
From your site collection you can also go to Site Settings > Term Store Management (same thing as in Central Admin).
There you will be able to edit all your term sets (not key words). Term sets belong to your taxonomy or folksonomy. You're actually talking about a folksonomy as your users are able to enter their own terms. Afterwards, as you said correctly, you probably want to clean up these terms by your users. Do try to use synonyms as well, so you have one main term as well as synonyms.