Chemistry of phosphodiester bond formation by DNA polymerase
I found this paper, which goes very deep into the molecular details of the individual steps of this reaction and also discusses how this is coupled to nucleotide selectivity.

The 'basic' details about the reaction (quoted from this section, which also has a nice figure):


  The polymerization reaction proceeds by a simple nucleophilic attack
  of the 3'OH of the primer on the α-phosphate of the incoming dNTP
  followed by the elimination of pyrophosphate [...] The reaction uses a
  "two metal ion" mechanism in which metal ion A activates the 3'OH as a
  metal hydroxide while both metals A and B stabilizes the developing
  negative charge on the α-phosphate in the transition state.


The metal ions are Magnesium (Mg$^{2+}$) and are properly positioned by the enzyme structure and some additional amino acid side chains also help with the activation of the reaction.