I look at the symbolism found throughout the series to create my own interpretation of the ending. when i first saw the series, i must have been around 12 or 13, and at that time my mind wasn't really able to grasp anything behind the huge mecha's and the action. but now that im older, from what I received from the series was that of lonliness, each indivdual main character is alone from the start, then in the middle they have eachother, but at the end they all fall alone again. through an emotion spiral and a psycological journey, each of the characters must find themselves, and in order to do this they must do it alone. example is that Shinji, being very weak minded, needs a crutch throughout the series, his crutch being EVA, but at the end, he's left to deal with his many trials and tribulations alone. that's what i received from it.
Strong bump, but whatever.
The interpretation I came up with was that Shinji wanted to be alone with no one else around him so that he could not be hurt or hurt others. Being hurt by others/ hurting others and the pain that brings is repeated many times in Eva. Shinji wishing to be alone is also repeated a lot.
When he awoke on the beach he saw Asuka there. Thinking that she would end up hurting him or that he might hurt her, he wanted to kill her to insure neither of those could happen. Then when she showed affection and kindness towards him, something he had never felt, something he had not expected, (plus all the other shit that had gone down must have been insane and emotionally painful... he DID end the world after all), he broke down and cried.
But another interpretation I have heard goes like this;
When Shinji is having his psycho mind****, there is a part where Asuka says (In the literal translation) "But, if I have to be with you, I'd rather die!". So when Shinji sees Asuka next to him what he is really doing is an act of mercy, killing her because (he thinks) she would rather die than be with him. It's an act of twisted love.
Asuka also wants this because there is no reason for her to live. Like in her previous suicide attempt, her raison d'être (piloting an eva) is taken away from her. She WANTS to die. However she is unable of killing herself as she is wounded and can only move one arm. When Shinji can't do it, she is disgusted (at how weak he is).
Shinji cries because Asuka's caress reminds him of his mothers farewell. (Who also caressed Shinji just prior to the beach scene)
I have heard more interpretations but these two are the ones I like the most.
My interpretation is that instrumentality has failed. Shinji stopped it. Rei appears to him so she can see him one more time. Since she is no longer physical. So in other words the fate of humanity rested on shinji. And he finally those a life of pain that involved the glories of everything he can experience such as love and lust over an escapist reality. As for the end with asuka I think that possibly Asuka has taken on some of Rei's spirit as evident from her bandages and now black eyes. One thing we are not sure of is wether or not the scene at the beginning is Asuka after teh final battle and instrumentality or her state before that. This as well is given to interpretation.
I think you can try for years to figure out the meaning of NGE, as though everything has some deep insightful significance and purpose, but the reality is that there isn't anything like that. The creator just thought stuff would look cool and liked using pseudo-Judeochristian nonsense for whatever reason. For a long time I thought that NGE had some hard to reach meaning, but it doesn't. It's just pure insanity and WTFs. That's the point of the ending. Either they ran out of money or deadlines came up, and they were like, "How are we going to end this," and someone was like, "Hey, let's go with a WTF ending; those always work."
Don't get me wrong. I like the show. But there's no reason to over analyze it.