What did Neo fight for?

One of the major themes of the Matrix trilogy is that of, essentially, predestination versus free will. The Oracle, while treated as being capable of seeing the future, states that she can't see past choices that people don't understand. She counsels Neo to "know thyself" in the first movie.

Many of the Machines, however, believe in as little choice as possible. From the Keymaker's "we do only what we are meant to do", to the Architect's creation of the Matrix to keep people's choices locked down. In the second movie, the Architect explains that the first Matrixes were "perfect" but lacking in choice, so humans rejected it.

The first Matrix I designed was quite naturally perfect, it was a work of art – flawless, sublime. A triumph equalled only by its monumental failure. The inevitability of its doom is apparent to me now as a consequence of the imperfection inherent in every human being. Thus, I redesigned it based on your history to more accurately reflect the varying grotesqueries of your nature. However, I was again frustrated by failure. I have since come to understand that the answer eluded me because it required a lesser mind, or perhaps a mind less bound by the parameters of perfection. Thus the answer was stumbled upon by another – an intuitive program, initially created to investigate certain aspects of the human psyche. If I am the father of the matrix, she would undoubtedly be its mother.

(Matrix Reloaded Transcript)

It took the Oracle to add choice into the Matrix so that people would subconsciously accept it as real. According to the Architect, the mathematical errors that choice causes are summed up in the One, and the One is used to rebalance his equation.

Smith, as an Agent, was originally bound by the rules of the system, even as he attempted to rebel against it by privately interrogating Morpheus for the codes to Zion. When 'destroyed' by Neo at the end of the first movie, it instead left him with something that allowed him to not be deleted, but he still doesn't think himself, or anyone, free.

Smith: I killed you, Mister Anderson, I watched you die… With a certain satisfaction, I might add, and then something happened. Something that I knew was impossible, but it happened anyway. You destroyed me, Mister Anderson. Afterward, I knew the rules, I understood what I was supposed to do but I didn’t. I couldn’t. I was compelled to stay, compelled to disobey. And now here I stand because of you, Mister Anderson, because of you I’m no longer an agent of the system, because of you I’ve changed – I’m unplugged – a new man, so to speak, like you, apparently free.

Neo: Congratulations.

Smith: Thank you. But as you well know, appearances can be deceiving, which brings me back to the reason why we’re here. We’re not here because we’re free, we’re here because we’re not free. There’s no escaping reason, no denying purpose – because as we both know, without purpose, we would not exist.

(Also Matrix Reloaded Transcript)

Smith also does things without understanding, he denies others the choice of what they want to be and do, and he spreads himself because he can. He is the virus he railed about humanity being in the first movie.

When he assimilates the Oracle in the third movie, he believes himself to have the power of infallible foresight, which would only be possible if there were no choices to be made. When he asks why Neo fights, all of his reasons are ones of purpose, and ones that could be interpreted as reasons of duty or obligation, rather than free will.

Neo's response to Smith is very simple, and a refutation of Smith's entire worldview: he made a choice. He could have decided to fight or not to fight, and he chose to fight. He wasn't artificially constrained into doing it. He could even have chosen to run away from both Zion and the Matrix. At the end of the fight, he chose to give himself up, which allowed Deus Ex Machina to purge Smith from the Matrix, and allowed the truce that he brokered with Deus Ex Machina to come to pass.

For himself, he chose to be able to choose, and he also chose to let the others in the Matrix make choices for themselves.