So you've just acquired a new rep-based privilege. There's a couple of different questions here:
What is required?
Nothing. These milestones are privileges, not responsibilities.
How should you use this power, given that you want to?
First of all, read the help centre page associated with that particular privilege. (There a list of all privileges with links to some explanation and guidance for each one, so this doesn't just go for 3k powers.) That page is the same on all Stack Exchange sites, so it contains nothing SFF-specific; it's mainly about the technicalities, system restrictions, and user interface associated with this power.
You'll notice that that page links to the on-topic page in the help centre. Again, the page exists on all Stack Exchange sites, but this one is tailored specifically to SFF (in fact, site moderators can edit it). The most important information to take away from there is:
Science Fiction & Fantasy Stack Exchange is for questions targeted
towards science fiction and fantasy enthusiasts. This includes
- Plot, character, or setting explanations
- Historical or societal context of a work
- Behind-the-scenes and fandom information
- Story identification
- Franchise/series reading or viewing order
Not all questions have a home on Stack Exchange. Please note the
following types of questions are off-topic here:
- Questions calling for a list of works, authors, …: What
are all the books that have X? Who wrote about topic
- Reading or viewing recommendations: I liked X,
what should I watch next?
- Genre classification: Is X
For more information, see the
question where these topics were discussed on our meta discussion
OK, so now what do you do when you come across a question in the Close Votes or Reopen Votes review queue and need to work out what to do with it? Well, there are a few different reasons a question might need to be closed:
Duplicate. These are relatively straightforward to work out. The review queue will show you two tabs, for the question you're voting on and the proposed duplicate. Read through the dupe-target and its answers; if the other (usually newer) question is answered there, then vote to Close/Leave Closed, otherwise vote to Leave Open/Reopen.
Unclear what you're asking. If the question is not actually a question, or it's so unclear that you can't understand what's being asked, vote to Close/Leave Closed.
Primarily opinion-based. If the question is not objectively answerable - e.g. it's asking for 'the best' something, or about future works - then vote to Close/Leave Closed.
Too broad. If the question has such a broad scope that any answer would necessarily be ridiculously long (OK, that's vague - for better guidance, see this meta question and answer), vote to Close/Leave Closed.
Off-topic. This is probably the hardest and most complicated to determine. There are a couple of standard off-topic close reasons - scientific solutions or explanations, explained in detail here, and lists of works or recommendations, which is fairly self-explanatory. Beyond that, it can be harder to determine. Sometimes it's obvious - questions which are obviously nothing to do with science fiction or fantasy - but more often it's not at all clear-cut and you may need to spend a while thinking about it.
Don't forget that if you're not sure - e.g. because the question is about a work you're not familiar with, or just because you don't want to commit yourself one way or the other - you can always click Skip and not review that particular question. There are also some friendly folk in chat who'll be happy to discuss or advise on question-closing dilemmas. And if you want a more formal discussion and official consensus, you can post a scope question here on meta, either about a specific-question or about a whole class of questions, asking whether they should be allowed or closed.
Also remember that if you do go wrong, it's not the end of the world. You can't close or reopen a question all by yourself (at least until you get a gold tag badge and the associated dupehammer), and any closed question can be reopened, any reopened question re-closed. Platitudes about great responsibility coming with great power don't apply that much to 3k rep users :-)