What is the difference between mentioned phrase and "I want to be the me"? Is it right to use both phrases in the spiritual growth context? I what parts we could deconstruct this phrase? Is the "I want to be" the attribute there?
"Me" is used in a very non-standard way here. It would often be stylized to show that it is unusual:
the "Me" I want to be
the Me I want to be
the me I want to be
Your rearrangement I want to be the me is not correct. It doesn't make sense, and additionally it changed the original which was not a sentence into a complete sentence.
To aid you in understanding the original meaning, the best suggestion I can offer is to replace me with person. It will probably fit into the same contexts:
the person I want to be
Does this make it easier to understand?
(I did not address your question about the "attribute," since I don't really know what that means.)