Making sense of The End of Evangelion
First: See my infodump on Schopenhauer, because this will in part assume you've read it. Also, my post on Kaworu & psychoanalysis will probably help to understand this too.
Second, this will be, in some ways, only a partial explanation. Evangelion is soaked in psychoanalytic theory, philosophy, and what I can only call postmodern Christianity.
Third - TW/CW: Discussion of suicide, and also THAT scene (IYKYK)
To understand the themes of the EOE, we need to begin with a primer. One philosophical idea Evangelion wears on it's sleeve is Schopenhauer's thought experiment: The hedgehog's dilemma. The idea goes like this - two hedgehogs (or porcupines) are in the freezing cold and wish to huddle together for warmth. However, either animal would pierce and kill the other if they got too close. This experiment is seen as a metaphor for intimacy, and depending on your read of Schopenhauerian pessimism, you get different answers. It is my contention that Anno solves the dilemma, but we'll get there.
The other thing we have to discuss is Shinji himself. Shinji is frequently self-serving throughout the series. This is not entirely without reason, as he is clearly being emotionally harmed by those around him. However, he also frequently runs away from his problems & obligations, and projects maternity (and the obligations of it) onto the female cast of the series.
This all, of course, culminates in the infamous hospital scene.
I won't go into detail about the scene due to Tumblr guidelines, but it's worth noting that Anno here takes a massive departure from how he originally portrayed Shinji. Whereas previously he represented Anno's ego, in the movie, he represents the audience, and the audience has the camera turned on them, so as to critique an attitude he finds common in otaku. It's worth noting that Anno's reason for this change was due to the hostility the original ending was met with, where the message was one of support for the audience & their wellbeing.
With Shinji covered (no pun intended), we can move straight to the Third Impact, and what it means.
During the Third Impact, two particular concepts are at play. The first is Schopenhauer's will to life, and the second is Freud's death drive. The death drive, in psychoanalysis, posits that at the base level of our subconscious, is a drive to cancel all other drives, a drive towards annihilation. This isn't simply a suicidal inclination (Freud notes that offing yourself would still fit into the drive of the pleasure principle), but a desire to kill, destroy, smash, etc; One can hear this during the song that plays during the Third Impact, "Komm Susser Tod (lit. "Come Sweet Death")":
It all returns to nothing
Tumbling down, tumbling down, tumbling down
It all returns to nothing
Letting me down, letting me down, letting me down
Shinji is on the verge of suicide, and is willing to take the entire planet with him. However, like anyone who's ever been on the ledge knows, dying isn't really the desired effect, just the best idea for ending pain. For Shinji to end the Third Impact without killing everyone - in other words, to step down from the ledge - he must solve the hedgehog's dilemma.
The dilemma is shown throughout the series with multiple characters, but none as thoroughly as Misato & Kaji, who I'd like to focus on here, because it will elucidate Anno's answer to the dilemma. Throughout the series, Misato tries to dictate the emotional position between her & Kaji. Not because she lacks, but precisely because she has feelings for him.
At first glance, this appears to be the reasonable answer to the hedgehog's dilemma - if closeness is dangerous, then the smart thing to do is dictate how close one is willing to be. But this solves nothing, and Misato's regret and frustration after Kaji's death shows us why: By trying to do this, she hasn't actually done anything. Rather, she's restated the dilemma in different terms.
So then, how does Shinji of all people manage to solve the hedgehog's dilemma? He is, after all, profoundly anti-social and irresponsible. In the end, Shinji is told that if he doesn't end the world, he'll still have his fear of the Other. And he accepts that. In doing so, in this moment, we see how Anno solves the dilemma: By dissolving it as such.
The hedgehog's dilemma, stated differently, could just as well be: "Accept the existence of other people or die." And a meaningful dilemma would not have a clear lesser of two evils. In the end, both by criticizing an audience that fails to care for itself, and by using Freud to overcome Schopenhauer, Anno's message is clear: Touch grass.