Dominick "Dom" Cobb (Leonardo DiCaprio) and Arthur (Joseph Gordon-Levitt) are "extractors", who perform corporate espionage using an experimental military technology to infiltrate the subconscious of their targets and extract valuable information through a shared dream world. Their latest target, Japanese businessman Saito (Ken Watanabe), reveals that he arranged their mission himself to test Cobb for a seemingly-impossible job: planting an idea in a person's subconscious, or "inception".
To break up the energy conglomerate of ailing competitor Maurice Fischer (Pete Postlethwaite), Saito wants Cobb to convince Fischer's son and heir, Robert (Cillian Murphy), to dissolve his father's company. In return, Saito promises to use his influence to clear Cobb of a murder charge, allowing Cobb to return home to his children. Cobb accepts the offer and assembles his team: Eames (Tom Hardy), a conman and identity forger; Yusuf (Dileep Rao), a chemist who concocts a powerful sedative for a stable "dream within a dream" strategy; and Ariadne (Ellen Page), an architecture student tasked with designing the labyrinth of the dream landscapes, recruited with the help of Cobb's father-in-law, Professor Stephen Miles (Michael Caine). While dream-sharing with Cobb, Ariadne learns his subconscious houses an invasive projection of his late wife Mal (Marion Cotillard).
When the elder Fischer dies in Sydney, Robert Fischer accompanies the body on a ten-hour flight back to Los Angeles, which the team (including Saito, who wants to verify their success) uses as an opportunity to sedate and take Fischer into a shared dream. At each dream level, the person generating the dream stays behind to set up a "kick" that will be used to awaken the other sleeping team members from the deeper dream level; to be successful, these kicks must occur simultaneously at each dream level, a fact complicated due to the nature of time which proceeds much faster in each successive level.
The first level is Yusuf's dream of a rainy Los Angeles. The team abducts Fischer, but they are attacked by armed projections from Fischer's subconscious, which has been trained to defend against extraction. The team takes Fischer and a wounded Saito to a warehouse, where Cobb reveals that while dying in the dream would normally wake Saito up, the powerful sedatives needed to stabilize the multi-level dream will instead send a dying dreamer into "limbo", a world of infinite subconscious from which escape is difficult and a dreamer risks forgetting they are in a dream. Despite these setbacks, the team continues with the mission.
Eames impersonates Fischer's godfather, Peter Browning (Tom Berenger), to suggest Fischer reconsider his father's will. Yusuf drives the van as the other dreamers are sedated into the second level.
In the second level, a hotel dreamed by Arthur, Cobb convinces Fischer that he has been kidnapped by Browning and Cobb is his subconscious protector. Cobb persuades him to go down another level to explore Browning's subconscious (in reality, it is a ruse to enter Fischer's).
The third level is a snowy mountain fortress dreamed by Eames. The team has to infiltrate it and hold off the guards as Cobb takes Fischer into the equivalent of his subconscious.
Yusuf, under pursuit by Fischer's projections in the first level, deliberately drives off a bridge and initiates his kick too soon. This removes the gravity of Arthur's level, forcing him to improvise a new kick that will synchronize with the van hitting the water, and causes an avalanche in Eames' level. Mal's projection emerges and kills Fischer, Cobb kills Mal, and Saito succumbs to his wounds; all three fall into Limbo. While Eames sets up a kick by rigging the fortress with explosives, Cobb and Ariadne enter Limbo to rescue Fischer and Saito.
Cobb reveals to Ariadne that he and Mal went to Limbo while experimenting with the dream-sharing technology. Sedated for a few hours of real time, they spent fifty years in dream time constructing a world from their shared memories. When Mal refused to return to reality, Cobb used a rudimentary form of inception by reactivating her totem (an object dreamers use to distinguish dreams from reality) and reminding her subconscious that their world was not real. However, when she woke up, Mal was still convinced that she was dreaming. In an attempt to "wake up" for real, Mal committed suicide and framed Cobb for her death to force him to do the same. Facing a murder charge, Cobb fled the U.S., leaving his children in the care of Professor Miles.
Through his confession, Cobb makes peace with his guilt over Mal's death. Ariadne kills Mal's projection and wakes Fischer up with a kick. Revived at the mountain fortress, Fischer enters a safe room to discover and accept the planted idea: a projection of his dying father telling him to be his own man. While Cobb remains in Limbo to search for Saito, the other team members ride the synchronized kicks back to reality. Cobb eventually finds an aged Saito in Limbo and reminds him of their agreement. The dreamers all awaken on the plane and Saito makes a phone call.
Upon arrival at Los Angeles Airport, Cobb passes the U.S. immigration checkpoint and Professor Miles accompanies him to his home. Cobb tests reality using his totem, a spinning top that spins indefinitely in a dream world, but ignores its result and instead joins his children in the garden.

Question: Where does Cobb find the opportunity to enter into the subconscious of Robert Fishcher?
Answer: On a plane flight to L.A.