The code exposes two public functions:

   Twitter_AuthenticateNewCredentials()
   Twitter_SendTweet(credential_bundle, status_text)

The first leads the user through the procedure to grant your application permission to send tweets on their behalf. It returns a “credential bundle” (a Lua table) that can be cached locally (such as in the plugin preferences — see LrPrefs) and used for sending subsequent tweets forever, or until the user your application at Twitter unpermissions.

For example, if you have TWITTER_CREDENTIALS in your exportPresetFields list (with its default set to nil) and P is the local copy of the property table for the plugin (e.g. as passed to sectionsForBottomOfDialog, you might have:

  f:view {
     bind_to_object = P,
     place = 'overlapping',
     fill_horizontal = 1,

     f:static_text {
        fill_horizontal = 1,
        visible = LrBinding.keyIsNotNil 'TWITTER_CREDENTIALS',
        LrView.bind {
           key = 'TWITTER_CREDENTIALS',
           transform = function(credentials)
                          return LOC("$$$/xxx=Authenticated to Twitter as @^1",
                                     credentials.screen_name)
                       end
        },
     },
     f:push_button {
        visible = LrBinding.keyIsNil 'TWITTER_CREDENTIALS',
        enabled = LrBinding.keyIsNotNil '_authenticating_at_twitter',
        title   = "Authenticate at Twitter",
        action  = function()
                     LrFunctionContext.postAsyncTaskWithContext("authenticate at twitter",
                        function(context)
                           context:addFailureHandler(function(status, error)
                                                        LrDialogs.message("INTERNAL ERROR", error, "critical")
                                                     end)
                           context:addCleanupHandler(function()
                                                        _authenticating_at_twitter = nil
                                                     end)
                           _authenticating_at_twitter = true
                           P.TWITTER_CREDENTIALS = Twitter_AuthenticateNewCredentials()
                        end)
                  end
     }
  }

and then later during export...

  local P = exportContext.propertyTable

  if P.TWITTER_CREDENTIALS then
     local result = Twitter_SendTweet(P.TWITTER_CREDENTIALS,
                                      "I just did something with Lightroom!")
     if result == nil then
        -- user has revoked permission, so we'll uncache the credential bundle
        P.TWITTER_CREDENTIALS = nil
     end
  end

You will have to update the code to have it reference the specific Consumer Key and Consumer Secret granted to your application when you registered it at Twitter.