It means "Based on his experiences as the greatest wandering philosopher of the Renaissance, he can confidently say (and we would do well to believe him) that it is important to see as much of life as possible but take time time to notice it too."
We can use "XXX should know" to indicate that the person knows it from experience.
Temperatures fell well below zero last night. I should know - I was outside all night!
Sarah says Milton Keynes is one of the most depressing places in the UK, and she should know - she lived there for 15 years.