Jesus as The Way is a metaphor, and profound one at that. The word "way" used in the New Testament is used properly as a road or a "path traveled", as in: "the three wise men traveled back a different way." But as you observed, Jesus is obviously not a literal road, or a path - he is not made of rocks or gravel, and you don't walk on him. The metaphorical use of this word however has many fantastic nuances, most importantly, picking up on the Old Testament use of the word "way" - as in its many references in Proverbs: "I have taught you in the way of wisdom" (Prov. 4:11), "Don't go in the way of evil" (4:14), etc.
To take this a bit further, notice that "wisdom" is often personified in some of these Old Testament passages (namely, Prov. 8). For the Hebrews, this was a way of embodying the very truth of God (embodying not being the best word, but the significance is that it wasn't just some esoteric idea or abstract truth; it was a personal, albeit metaphorical, reality). Jesus in a sense is claiming to be that very embodied word of God, as John 1 and other places testify. Following Wisdom's way leads to eternal life - not merely "going to heaven when you die" but the Kingdom of God - the rule and reign of God that will one day fully manifest in the "New Heavens and New Earth", but is also dramatically breaking-in here and now with the arrival of Jesus.
One helpful passage for starters might be Mark 12:28-34, where Jesus instructs the scribe to "Love the Lord your God with all your heart, soul, mind, and strength...and love your neighbor as yourself," and upon seeing that the scribe understands, tells him, "you are not far from the Kingdom of God." This is following the Way, the Jesus Way. There are many other ways one can follow: the way of putting yourself first, the way of the sword, the way of materialism and greed, but none of these ways are truly life giving, none lead to Eternal Life. Only the way of Jesus does.
It is also interesting to note that the first Christians were called followers of "The Way" (Acts 19:23, for example). This I think is representative of the earthiness of following Jesus - it was a "way of being" in the world: there are literal acts of love and kindness involved. It is not simply a matter of "believing the right things" - though that is important. Faithfulness to Jesus involves discipleship, apprenticing yourself in all aspects of life to the Master, the One who is the source of all life and truth.
This has nothing to do with a so-called "works-righteousness" - that is simply missing the point. This is about what it means to actually live in the world between Resurrection and Judgment Day, what it means to live in the Kingdom of God "on earth as it is in heaven."