'A lion does not turn around when a small dog barks' means:
Small problems are insignificant.
Ignore minor distractions.
Barking dogs can hurt lions.
Only lions can ignore dogs.
'A person who has not traveled widely will think his/her mother is the best cook' means:
One can never know the world without leaving home.
Home cooking is the best.
Cooking is a cultural experience.
Traveling opens one's eyes to new experiences.