River is a song of repentance and of hope. Leon realizes his past and how he hasn't lived the life he should. It says
"Tip me in your smooth waters
I go in
As a man with many crimes
Come up for air
As my sins flow down the Jordan"
Meaning that whe he gets baptized he comes out a new man and his past is now behind him. By "traveling the wide roads" he was going down the path wide, evil path instead of the straight yet narrow path that leads to righteousness. The song is about becoming pure and living a life of repentance from evil ways.