This is the 12912th most frequent Russian word.
лесному
"лесному" means "to the forest" or "forest-related" (dative case, singular, masculine).
In this sentence, 'лесному' is used as a dative singular form of the adjective 'лесной', meaning 'forest', modifying 'проселок' (trail), denoting location and describing its characteristic as related to the forest.
Мы отправились в путь по лесному проселку.
We set off on the forest trail.
Here, 'лесному' modifies 'хозяйству' (management) in the dative singular case, referring to management related to forests.
Она подарила книгу о лесном хозяйстве.
She gave a book about forest management.
In this example, 'лесному' describes the shore ('берегу') in the dative singular case, indicating a connection to a forest by specifying the shore's nature.