Singing lets you jump over the fences of your life. Writing lets me jump over the fences of my life.
Singing lets you jump over the fences of your life [in] the way [that] writing lets me jump over the fences of my life.
There is a lot of ellipsis in the original sentence which my examples above might help clarify. "Singing" and "writing" are the subjects of their respective clauses. The predicates of those clauses are nearly identical -- same verb, same infinitive phrase. The differences are objects -- "you" and "your life" in the first, "me" and "my life" in the second.
To avoid unnecessary repetition, we'll keep only the elements of the second clause that are different. We'll keep the subject "writing". We'll replace the verb "lets" with "does". The rest we'll lose to ellipsis. This gives us a second clause that's almost correct:
- [that] writing does [let you jump over the fences of your life]
We need some way to change the references in the objects, because we want this clause to mean:
- [that] writing does [let
you me jump over the fences of your my life]
We do that by adding "for me" to the elliptical clause:
- [that] writing does for me
Even after all these omissions and revisions are made, we're still left with a relative clause which modifies "the way". The subject remains "writing". The verb "does" stands for the entire predicate of the original clause. The prepositional phrase "for me" modifies the verb "does", in effect changing the objects of the omitted predicate.
In turn, "the way" acts as if it's the object of a preposition. The construction "the way writing does for me" modifies the clause "singing lets you jump over the fences of your life." Everything from "that singing" to the end of the sentence is a relative clause that modifies "feeling", which itself is the direct object of the verb "got".