The phrase "Swan geese are busy heading north to their hometown" means that swan geese fly north in autumn and return to the hometown where they were born.
This is a natural phenomenon and a symbol of seasonal changes. Swan geese are migratory birds that migrate every year as the seasons change. In autumn, they fly from the south to the north, returning to their birthplace. This sentence may express people's longing for their hometown and their attachment to their hometown.