Most sentences in this song don't match the lines, so it will be much easier to understand if we reconstruct the complete sentence where necessary.

IS NOT THE WORDS (THAT) I WANT...= The subject of this sentence is the previous line: "saying that I love you". So the complete sentence is: "Saying that I love you is not the words I want to hear from you".

IT'S NOT THAT I WANT YOU NOT TO SAY= It's not that I don't want you to say "I love you", that's not the problem.

BUT IF YOU ONLY KNEW HOW EASY IT WOULD BE TO SHOW ME HOW YOU FEEL= ONLY here is just emphasizing, but the expression IF ONLY is often used in English to express a wish, followed by the past (or "would") (e.g: If only you were here / If only she would love me.). So here, he wishes she knew.

MORE THAN WORDS IS ALL YOU HAVE TO DO TO MAKE IT REAL= If we rearrange the order of this sentence it would be:
(purpose:) In order to make it real, (subject:) all you have to do (verb:) is (complement:) more than words.
Now we can see that the action is very strange, because "more than words" is not a verb, is not something you can do. But this is poetry and we have to interpret it. The meaning would be:
To prove your love, you need to do more than just saying nice words (you have to do things to show it).
In the third paragraph he'll explain what things she can do to show her love without words (all you have to do is close your eyes and just reach out...)

'COZ I'D ALREADY KNOW= Because I would know already.

IF MY HEART WAS TORN IN TWO= The verb TORN is the past participle of TEAR (tear-tore-torn), which means BREAK. The difference is that BREAK may be a clean cut, but TEAR is more messy, rough. You can break a sheet of paper by cutting it with scissors, but if you tear it, you use your hands, and the result is more irregular. So if your heart is torn, we get the idea that it is more painful than if we just say that your heart is broken. IN TWO, because when you break it you get two parts.

MORE THAN WORDS TO SHOW YOU FEEL THAT YOUR LOVE FOR ME IS REAL= The beginning of the sentence would be, for example, "You need more than words...".

WHAT WOULD YOU SAY IF I TOOK THOSE WORDS AWAY?...= If I could eliminate the words "I love you" (and similar) from your vocabulary, what else could you tell me to show your love?

NOW THAT I'VE TRIED TO TALK...= In this paragraph he's going to give an example of how she can show her love without words.

HOLD ME CLOSE= The verb HOLD, with people, means "hug, embrace". The word CLOSE is not a verb, but a preposition, so it means "near". If you hold someone close, you hug them tightly, strongly.

LET ME GO= This sentence could literally mean "allow me to leave", but most of the times (including here) it is an expression that means "stop holding me, set me free". In the song they use it in the negative, so "don't let me go" means "keep holding me".

DON'T EVER LET ME GO= Never let me go, keep holding me forever. To avoid a double negative, either we use NEVER with an affirmative verb or we use EVER with a negative verb (DON'T).