We never get to see the face of the old aunt living upstairs, in the flat above Ila's. All we get, throughout the movie, is her VOICE. The Lunchbox is all about communication.
Saajan Fernandes (Irrfan Khan) is an extreme loner who has never really communicated with people after his wife's death. He just does his achingly boring job and goes back to his empty apartment day after day. Ila (Nimrat Kaur) is a lonely housewife whose husband is always out at work (and possibly cheating on her). When he is at home he never really talks to her (doesn't even look at her properly). The old aunt living upstairs is a lonely old woman whose husband cannot talk (Ila tells us he just keeps staring at the ceiling fan). They are all lonely souls who, whether they know it or not, feel the terrible need to communicate.
So they talk to each other. It doesn't matter the fact that they are strangers and not really related to each other in any significant way. It doesn't matter that they are talking about something utterly banal and not something important or interesting. It doesn't matter that they are talking to the wrong person. What matters is that they talk, and that their talk be honest. The misplaced lunchbox and the food just become an excuse for two lonely strangers to communicate with each other. An excuse for sending letters to each other (Hence, the allusion to Sanjay Dutt's Saajan). And, the love that occurs between them is not a physical love. They never meet! It's a love that's developed out of pure communication.
Saajan isn't a particularly likable person. He's reticent and rude. He isn't friendly with others. He isn't kind to children. He isn't helpful. But there is one thing good about him: He is very honest. He's real. Whatever he says, he means it. There is no falseness in him. The first time Saajan replies to Ila, he, instead of thanking her, tells her that the food was salty. It is perhaps rude, but, at least, it's honest - unlike her husband's praises. He only lies at one point in the movie: when he tells his boss that he had made all errors in the file, and not Shaikh. But that becomes such a beautiful lie, because, for the first time, he explicitly shows KINDNESS towards another person. He lies not in order to deceive or cheat, but in order to save another person.
Shaikh (Nawazuddin Siddiqui), on the other hand, is extremely talkative. He talks too much and all the time. There is a childish innocence about him. He is curious and enthusiastic about everything. His personality wavers between being annoying and charming. In that sense, he couldn't be any different from Saajan. Saajan is very taciturn, jaded, and indifferent. And yet he is similar to Saajan in one fundamental way: He is extremely honest, too. His honesty exudes through his voice, his eyes, his body-language. He lies and cons here and there, but he does that all out of necessity and for a greater good, and never with the intention of hurting or deceiving another person. Plus, despite having a good wife, he's also a loner. He does not really have his own people around him. At his wedding, the only person by his side is Saajan!
All these people Sajjan meets become an incentive for Saajan to learn to love human beings once again. This movie charts Saajan's transformation from being a misanthropic loner to being a person who is ready accept people in his life. He tells us that he is looking forward to go to Nashik after his retirement. But, by the end, he doesn't go there. Nashik, according to Wikipedia, is said to be the place where Lord Rama spent his 14 year exile. So, in other words, Saajan doesn't go to exile. He decides to live among the people. The last shot of him sitting in a train surrounded by a group of people singing and clapping.