What does Connect.js methodOverride do?

If you want to simulate DELETE and PUT, methodOverride is for that.
If you pass in the _method post parameter set to 'delete' or 'put', then you can use app.delete and app.put in Express instead of using app.post all the time (thus more descriptive, verbose):

Backend:

// the app
app.put('/users/:id', function (req, res, next) {
  // edit your user here
});

Client logic:

// client side must be..
<form> ...
  <input type="hidden" name="_method" value="put" />
</form>

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