To clarify my understanding of the OP's question, I take it is meant by story the story of Lawrence, during his time there for the period covered by the film, and to be even more precise the film's version of his real story (I have read about that separately).
The story as told by the film is not merely a sequence of events, but follows a certain narrative arc. ON the level of the obvious this includes shifting back and forth between his dealings with the British Army and his presence in the desert with the Arab Revolt, as well as two scenes with Allenby and Dryden that Prince Faisal is also present. And the scenes in Damascus.
The shifting underscored the themes of the story, which for me is an examination of who Lawrence was, and how his environment and settings affected the answer to that question.
To be sure it is not merely a story that shows the effect of environment and events on a protagonist, because his own intentions also move the story forward. But that is the dynamic.
So, the answer I think is that what makes the story appealing is we see a man who has a mix of qualities, but who not only does not seem to fit in, and is uncomfortable with the situation he finds himself in. He is not being used, another way of saying he does not have the opportunity to engage himself better in his situation. He then is given an opportunity to take a different approach, but here by also going into an alien culture yet one he not only feels more comfortable in, but begins to succeed at.
The story then takes off with the Aqaba expedition and success. Here we see a dynamic combination of analysis, intention, courage and sheer physical effort lead to success. Lawrence's subsequent return to Cairo is one of the most interesting parts of the story. Success leads his fellow soldiers to see him in a quite different light, despite his "damned ridiculous" dress and appearance, taking on the visual trappings of "the wogs". By this time the perhaps main theme of the story is evident - who is Lawrence? And by extension who are we? Are we (as he later directly addresses) who we are in terms of some categorization of where we come from, what national or ethnic group we belong to? Or can Lawrence, and we, "write" our own story? Meaning find meaning in life not limited to those categorical imperatives?
Or can we either through choice or happenstance become someone else, by experiencing something else? How does what we do change who we were until that point, until we experience something new and different?
Where the real beauty of the story comes in is that the balance of it in effect says that the answer to this question is not so clear in Lawrence's case. Winning the race to Damascus may have not achieved the result sought, but on at least one other level Lawrence did achieve greatness and fame, as clearly shown in the second scene, with him memorialized in St. Paul's. Still the film's last scene leaves us with an answer that I think clearly shows Lawrence is neither the man of the desert warrior, "almost" arab, and is at the least not part of Allenby's army and what came with it. He is riding home to be sure, but to what? And what will his life in Arabia and what it meant mean when he returns to England?
In short the story is a fascinating examination of the pull between who we are in terms of where we come from and our experiences and their effects as we then go through new experiences that change us, and allow for expression of our intentionality. It is an Existentialist story at least to that extent. We are as we exist, but existence itself is a dynamic where the past and present of our lives and experiences mix. this speaks I think to all of us, or can if one cares to consider it.
That is why it is such a great story.
Despite the length of my previous post, I think it was inadequate. I left out the whole aspect of the other characters in the film being part of the story, for one. Their interplay was simply brilliant, and each of the supporting characters served a role as a sort of comparator to Lawrence. All were well done and added to the story, although I think Omar Sharif's Ali is my personal favorite. But Hawkins as Allenby and Raines as Dryden were also key to the story.
Another aspect I left out is the rather unique way in which our hero and protagonist was so deeply troubled both psychologically and morally. Hard to think of another character like that. Maybe Bogart in In a Lonely Place (was he really the hero there, though)? Monty Clift in From Here to Eternity (not really the protagonist)? John Wayne in The Searchers was certainly a very complex character psychologically, but he shared none of O'Toole's portrayal's moral ambivalence. Even films coming after LoA are hard to find examples of this kind of character and his story. Maybe Eastwood in Unforgiven. DeNiro in Taxi Driver? Hah! Too much the anti-hero. No, Lawrence's character made the story unique. It is compelling the way such a troubled person fought through his demons and did what he did. I think that is important as well.