Why is the 'You talking to me' speech from the movie 'Taxi Driver' so famous?

Just to get the more superficial reason out of the way first, it's possible that those particular words got stuck in our heads and are now representative of this entire scene because they're repeated 4 times during his speech. Repetition simply drills certain things into our heads. But beyond that, I think this is mainly about emotional complexity and intimacy. Our protagonist is unravelling and this is portrayed through a monologue (or soliloquy) into a mirror where he unknowingly shares his emotions with us. This device harks all the way back to Hamlet, but here, it's imbued with realism through Bickle's psyche and demeanor that De Niro (who improvised the lines) conveys in a chillingly natural way.

The scene is also oddly playful, putting us in Bickle's shoes (what we see is his reflection) as we play make-believe with him, maybe even triggering memories where we imagined standing up to someone, or practised speeches in front of a mirror - but contrasting it with his fractured mind and violent plans. This establishes a connection between us and him but also a repulsion, given the dangerous context; We also feel sorry for him: Not only does he seem delusional, but "I'm the only one here" also tells us just how lonely he is.

In short, I think it's memorable because it's a simple, short and naturally delivered phrase that actually represents a scene and character with a surprising amount of complexity. It's relatable but also scary and deeply troubling, which is why it ended up encapsulating Bickle's state of mind for us.

Why is it famous? The delivery.

You have to remember that this was entirely unscripted. It was the last week of shooting. They were filming some filler stuff. Scorsese just told DeNiro to just stand in front of the mirror and talk to himself. DeNiro, deep in his character, unleashed an inner view into the psyche of Travis, and it was instant gold. It's really the ultimate badass line.

"You talkin' to me? You talkin' to ME?!?"

That is a great question. I believe Marty McFly mimics De Niro's acting in back to the future 3, like you said "people either mimic Robert De Niro[...]". Besides the fact that the scene had some humor to it and was well played by De Niro, it is an important scene for the understanding of Travis' mental condition.

Have you noticed how he enjoys looking at himself in that mirror, mimicking a confrontation with possibly a thief? You see, Travis did not start buying guns to "sweep the city clean", as he claims and also tells the mayor candidate he should do. The feeling I have is that he enjoys violence, a near sociopathic behaviour, probably developed during his time in the Vietnam War (or maybe the reason he enlisted in first place).

Conclusion: his self-admiration in the mirror whilst holding a gun could symbolize his lust for violence, and his belief that he is better than all that "scum" in a "dirty" city.

He is seen as a hero by society for his deed. However, although he did something good, it was for the wrong reasons. Kind of like Dexter, in a way.

I think it is very clear. Let's see the scene

If DeNiro was holding a gun when saying the line, it would mean that the other had an opportunity to realize how dangerous DeNiro was, surrender and plead for his life.

As we are seeing, DeNiro is using the line without showing how ready he is to shot (not only he has a weapon, but also the contraption to hold it and shot at his target in a fraction of a second).

He is having a phantasy about how he is addressing someone, creating or escalating an argument and then killing his target "justifiably" without ever giving the victim any real option, not even the time for surrender. Next time someone tries to push him or does something he does not like, he will push back. And he likes that power a lot.

What is the reason behind this dialogue being so famous: 'You talking to me'?

1. The performance is iconic and memorable
2. Robert De Niro is a legend in film and his works are more remembered than lesser actors, and across greater numbers of cultures
3. the line is repeated, but with differing intensity allowing for it to enter into more aspects of society in multiple ways
4. the scene is visibly unique (or at least rare), making it more memorable
5. it is short and easy to remember
6. this phase is fairly common in speech and in circumstance, so it offers the opportunity to engage this line

I think it's important to note that this line is very famous among people who have never seen the movie. Not to take anything away from DeNiro's performance per the many answers. Surely that got the ball rolling, but there is something additional going on here. Something about that line transcends the scene where it originated.

Maybe because the line is short enough to remember easily and captures a feeling we all have at some time and didn't yet have the right words for.

I have heard this a million times, probably used it myself, and never saw Taxi Driver. Same with "I'm walking here!" from Midnight Cowboy, and other lines in @John's answer.

Certain phrases evoke an emotional response that comes from your gut. In doing so, an entire scenario appears in your head. What is it that makes "Are you talking to me?!" so powerful:

• It invokes the Fight or Flight instinctive response (slightly). Breathing, heart rate, blood pressure and more all elevate. So the phrase gets your "blood pumping".
• You are choosing to fight as an answer to the instinct. That means you are going to assert yourself in a manner that makes what you want actually happen.
• We all have had experiences where, afterward, we wish in some way that we would/should/could have done something like this. So it fits us all in way that is very personal to us.

Other phrases that come to mind that have this affect on us:

• Where's the beef?!
• What we've got here is failure to communicate!
• There are others that I invite the readers to add here.

(I was laughing as I wrote these. To this day they affect me.)

"You talking to me?" is an archetype that has relevance from caveman to now. The emotion of De Niro might have drilled it into our head, as far as the movie association but the question he is asking is a question every human has asked from 1 year old to death.

"You talking to me?" can range from the basic question to an emotional rage. So not only is the phrase also embedded in our head before watching, the movie allows us to invoke more freedom in how we use it.

"I know you are, but what am I? I know you are, but what am I?"

Maybe people just like saying 'the exact text of famous quotes.'