How is the type of genetic manipulation determined in CRISPR-Cas9?
But how do you get from double-strand breaks to editing genes in
  incredibly specific ways?


You should read the wikipedia page on CRISPR-Cas system. The target specificity is provided by the tracrRNA. 


  How do you specify which kind of repair mechanism is invoked?


The repair can be done either by Non Homologous End Joining (NHEJ) or Homologous Recombination (HR). If you provide excessive DNA template by transfection then it is likely to increase the chance of HR and this can be used for trangenesis. 

When no template is provided then DNA can be repaired by HR with sister chromatin or NHEJ. Depending on the stage of the cell cycle one of these mechanisms is preferred over the other. HR is shown to be the most active in S-phase and starts declining in G2-phase. 

So when you want to make transgenics or specific edits then co-transfect CRISPR-Cas with transgene DNA (with flanks homologous to target DNA sequence) or a homologous DNA with point mutation(s), preferably in cells at S-phase. It is possible to synchronize the cells using nocodazole or other molecules that arrest cell cycle.