あなた means “you” used toward someone in a lower status or equal in a polite manner. In this scene, Rei is speaking to Shinji, a classmate and coworker as Eva pilots for NERV.
は indicates that she is going to talk about あなた.
死なない means “won’t die”.
わ adds feminine flavour to the sentence which is often used in anime. In a real conversation, it can be replaced with よ which sounds more convincing. If you want to make a statement more declarative and decisive, you can just say あなたは死なない without any sentence ending particles.
私 means “I” commonly used by females, but in a formal situation, males use 私 as well.
守る means “protect”.
が is used when the speaker emphasises what comes before, which is 私 here. So Rei is emphasising that SHE is the one who will protect Shinji.
もの meaning “because” is used to give a good reason for what you do or what you say. It’s usually used by females or children. When you hear もの you will probably think of 物 meaning “a thing” or 者 meaning “a person” but this もの is a conjunction so It’s always written in hiragana. In colloquial speech, it’s often simplified to もん which sounds more childish. Here are some examples.
The clause with もの can be first followed by the main clause so 私が守るもの、あなたは死なないわ。meaning “I will protect you so you won’t die”. It depends on which sentence you want to emphasise. With the Japanese grammar structure, the important information comes at the end of the sentence so in this case, the emphasis is on the second clause “YOU WON’T DIE”.
So あなたは死なないわ、私が守るもの means “You won’t die, BECAUSE I WILL PROTECT YOU”. Rei is indicating her strong will that she is going to protect Shinji.