So this is an interesting translation, because what is actually being said, and the translation do not have the same literal meaning, but they carry the same general meaning as a figure of speech.
In short, a Japanese idiom is being translated to an English idiom.
Literally translates to:
I crossed over the mountain.
山 = mountain
は = topic marker. As a standalone sentence, I would use を instead, but you can have this be grammatically correct if this comes in the middle of a conversation. There are cases where は can replace を in a sentence and this is one of them.
越えた = past tense of 越える, which as you have found can mean 'to exceed,' but as @chocolate has pointed out, this definition uses a different kanji: 超える. It is read the same way. 越える, as used in this example, means 'to cross over/through.'
*As a side note, jisho.org is one of the best online dictionaries I have found, and I highly recommend it.
よ = sentence ending particle that adds emphasis (almost like an exclamation point, but not quite).
In terms of general meaning, the phrase means that you have gotten past some trying challenge in your life. The meaning is very similar to the idiom that it was translated to (I'm out of the woods), which is why I think that they used the English idiom over the literal Japanese translation.