If I may add more details, 眠い is "sleepy", not "tired". For "tired", there is 疲れる (tsukareru). The 目 radical in 眠 makes it clear since it means "eye". That is, 眠い has to do with the eyes (sleepiness) than with the body (tired).
"Because I am already tired" would be "もう 疲れた ので" (-た means state of being tired, or the past tense).
So the sentence means "As I am already sleepy, I think I will go to bed for the day".
Note that 眠い(nemui) is sleepy, while 眠る(nemuru) is the verb for "sleep" or "fall asleep". They use the same kanji.
Also, related to 眠る is the verb 寝る (neru) meaning "go to bed", which appears in the main clause.
眠る and 寝る might sound similar in meaning, but 寝る is more for the (conscious) act of going to bed ("turning in"), while 眠る is "falling asleep" (natural phenomenon). So they are sometimes but not always interchangeable.
In addition, the sentence has ので (because...) and ようと (intending to...) which are widely used important constructs.
Thank you all, it's more easy when it's explain :D
もう - it normally means "already", but here I think it emphasizes the state of being tired. Like Yutaka said, it kind of translates to very.
眠い - Tired (i-adjective)
ので - Because (conjunction)
Because I'm (really, very) tired.
Because I am very sleepy ...