Make sure you first complete the "Replace the Cow-Catcher" task. This becomes available while you are fixing up your train.

‼️​ Players who are already further into their adventure can travel back to the the Old Rail Yard first.
The Birdle game mode is unlocked by playing the "Unlock Birdle Quest" and finishing the task "Retrieve the envelope".
Be sure to keep playing to complete this task to unlock the Birdle game mode.

Once unlocked you can find the Birdle game mode in the main menu, at the left center side of your screen.