I know that 'ありのままで' means something like 'as I am', but how would one go about understanding the bits and pieces of the phrase, such as 'あり' and what its function in the phrase is? Thank you for your help.
- Ari あり: The masu-stem of ある ("to exist", "to be"). The masu-stem of a verb sometimes works as a noun. So here あり means "(your) being", "current existence", etc.
- No の: The particle that connects two nouns. You must be familiar with this.
- Mama まま: A noun that means "the status like before", "the same manner", etc. It's almost always modified by another clause or attributive. See: Learn JLPT N3 Grammar: まま (mama)
- De で: A particle that marks a condition. "in ～"
So they amount to something like "in the same manner as you are (now)" or "as (it) is". Basically ありのまま is a set phrase that works as a single no- or na-adjective.
Why "ari no mama de" is something like "as I am"?
ari no mama de is ありのままで. ありのまま is a set phrase which means そのまま, so ありのままで is そのままで or "as it is" or "Let it be".
I heard that ありのままで is translated for "Let it go", not for "as I am", "as it is" or "Let it be".
There is a story about the translation of 「ありのままで」.
The possible translation for "Let it go" is not "ありのままで" which is equivalent to "Let it be", but 「もう忘れたら？」,「もう気にするのやめたら？」 or 「もう手放したら？」.
But the translator thought the movement of the mouth of "Let it go, Let it go" is similar to that of "ありの～、ままで～" when someone sings the phrases of the lyrics. So he/she decided to translate it as it is, even if there might be a slight difference in the meaning.