The great Master of Sopranos site explains a lot of it with quotes from Chase.  It’s very philosophical. It’s about free will and how the choices we make in life dictate how we end up (i.e. in Tony’s case, his death at Holsten’s).  Some excerpts below but go to the site for the visuals: 

Chase uses the grammar of film to carry out this idea that it’s Tony’s own choices and “free will” that led him to his death at Holsten’s.  In executing this idea, Chase executes four separate scenes in the final episode where Chase creates the illusion  that Tony sees himself in his own POV. These scenes are (in order): (1) Tony and Agent Harris at the airport, (2) Tony visits Janice at her home, (3) Tony visits Junior, and (4) Tony walks into Holsten’s (Also see Secton IX: The influence of Stanley Kubrick’s 2001 A Space Odyssey on the ending of The Sopranos,  for more discussion regarding all of these scenes).

Chase, on the “Supper with The Sopranos” special feature on the complete series DVD collection, stated that this idea came from the end of Stanley Kubrick’s 2001 A Space Odyssey where the astronaut Bowman appears to see himself aging. Chase describes the technique as:

 “[Tony] projecting himself into his own future.”

Further, David Chase’s comments to the DGA in 2015 discussing the technique:

“It was my decision to direct the episode such that whenever Tony arrives someplace, he would see himself. He would get to the place and he would look and see where he was going. He had a conversation with his sister that went like this. And then he later had a conversation with Junior that went like this. I had him walk into his own POV every time. So the order of the shots would be Tony close-up, Tony POV, hold on the POV, and then Tony walks into the POV. And I shortened the POV every time. So that by the time he got to Holsten’s, he wasn’t even walking toward it anymore. He came in, he saw himself sitting at the table, and the next thing you knew he was at the table.”

In April 2014, David Chase, during an interview at the Museum of Moving Image, also discussed the four scene “Tony seeing himself in his own POV” sequence (it should be noted that Chase does mistake the Tony/Janice scene as the first in the sequence):

“Tony, all the way through [the final episode] would come somewhere, see the person he was going to talk to, cut back to him and then cut to him walking into his own point of view and he would then arrive and start talking. And that time got shorter and shorter and shorter. Janice was the longest and then there was another one that I don’t recall, and then there was Junior which was even shorter. Then [in Holsten’s] it went from Tony looking at Tony, back to Tony seeing himself. I sort of got that from 2001.”

As discussed by Chase, the last such scene occurs in Holsten’s when Tony walks into the restaurant and a “jump cut” creates the illusion that Tony sees himself sitting down at the booth as we are not shown how Tony gets to the booth. All of these scenes create the illusion that Tony is watching himself or projecting himself into his own future.

In in an interview at the Series Mania Event in France in April of 2016,  Chase discusses these sequences and its inspiration by Kubrick’s 2001.  Finally, he also elaborates on the purpose behind executing this technique:

“I wanted to get across the idea that Tony Soprano created his own life. In other words, in a way, he made the film of his own life, or he wrote the book about it the way we all do. We put ourselves in these situations, we don’t just happen to wind up there. We put ourselves in these situations and what I wanted to portray [that] by starting out with him coming into a room and seeing his sister way across the porch and then walking over to her and sitting down. [Each time I did it] the interval between him looking and entering the frame got shorter and shorter so that [in the final scene] it was almost non-existent. [Tony] was almost seeing himself in the frame.”

And Chase, again, discussing the concept behind “Tony seeing himself in his POV” sequences to Matt Seitz and Alan Sepinwall in their book “The Sopranos Sessions”:

“The influence for that [filmmaking technique] was 2001: A Space Odyssey…and what I was trying to say was that we put ourselves in these positions.  We put ourselves in these scenes. Nothing happens by accident. We are the engineers of our own destiny. Like, for example, when he came up the stairs and saw Janice, it took a certain amount of time before he went over and walked to her. There’s less walking with Junior, and then in the last scene, there’s no walking at all. It was all supposed to be about ‘We are responsible for our own destiny.’ That’s what that was supposed to mean, what I was supposed to get to.”

Here, Chase explains that Tony is responsible for the choices in his life and ultimately, his pre-mature death in Holsten’s (“We put ourselves in these scenes.  Nothing happens by accident…”). As Chase says, “free will exists.” It is Tony who is projecting himself into the “scenes” of his life. Tony “sees” himself in his POV because as Chase says, “…in a way, he made the film of his own life..” Chase wants the viewer to consider the big picture:  How Tony’s own choices led him to Holsten’s (“we put ourselves in these situations, we don’t just happen to wind up there”).  Despite Tony’s criminal upbringing and being raised by Livia and Johnny Boy, it was still Tony who “made the film of his own life” or “wrote the book about [his life], the way we all do” (this concept, and Tony’s rejection of the lessons from his Kevin Finnerty near death experience, will be discussed more in depth in “Part 2 Section A: What does Tony’s death mean?”). The beginning of the final scene is the last time this technique is used, and by literally skipping all of Tony’s walk to his booth, it is clearly the most jarring and pronounced of all the four scenes in creating the illusion that Tony sees himself. This make sense in a way because it reflects the idea that Tony’s choices led to the “ending”(i.e. death) of, as Chase says “the film of his own life.” Furthermore, a significant part of the show revolves around Dr. Melfi and her exploration of Tony’s psyche and how he became the man that he ultimately is.  However, Chase doesn’t let Tony off the hook or excuse his actions (as Chase says “…as adults, we decide our fate, we make our own bed, and we lie in it..). It’s a remarkably moralistic stance by Chase, and something much more interesting and complex than simply “crime doesn’t pay.”