In Inception when every one wakes up in the airplane, Saito finally makes the call so I can understand he knows Cobb successfully planted the idea in Robert's mind. But I don't understand why Robert doesn't know that someone else has planted the idea in his mind. Robert and Saito both go to three levels deep and get lost in limbo, so how does Robert not know that someone played with his mind?
A dream in the movie has the same features as any regular dream. When the unsuspecting subject wakes up, he has a vague memory of the events of the dream, whilst some personal faces can be recollected, the rest will be be a blur. This is just like any dream we tend to have and try to recollect it in the morning once we've woken up, some things would be clear, other events and faces would blur out. We will pass it off as a dream and continue with our morning routine.
This is the same thing that happens to Fischer, though he'd wake up thinking - "whoa, that was a crazy dream there, must be what I ate before the flight".
Why does Fischer accept the Inception? For Fischer, his whole life he wanted his father to appreciate him for his own skills. The toy Wind Mill is an object that represents Fischer's own creation. Inception of this object in his mind's safe lets Fischer to believe that his father always wanted him to be his own man and not merely run his father's company. This is something Fischer always wanted and so a small idea planted in his head grows and he finally disbands his father's company.
The dream, from Robert's perspective, seems very chaotic - just like any other dream. What is more, people sometimes dream of things they see while awake - the people around him. What is more, as Napoleon Wilson said in his comment, after waking up, the idea of the dream fades away.
Another argument in favor of the acceptance is that he was flying some public plane, probably owned by a well-respected company. Hence the mere thought that someone would be able to slip him some drugs and then connect him to the machine for a theft is pretty hard to believe (sounds very paranoid).
What is more, the inception is some sort of myth in that universe (remember the sequence in the beginning, where people tell Saito he's nuts, but it comes as a shock for everyone when Cobb admits to have accomplished an inception once).
There's a few things going on here that would make it unlikely for him to detect what's happened. For clarity, all quotes will come from from the shooting script.
Firstly (and most importantly) inception is supposed to be impossible. No matter how expertly done, when someone puts an idea into your mind, we're told that it's detectable. Fischer wouldn't be expecting it, any more than he'd be expecting his dead father to get up and start dancing:
SAITO: You could plant it subconsciously—
ARTHUR: The subject’s mind can always trace the genesis of the idea. True inspiration is impossible to fake.
Secondly, Robert Fischer is confident that no-one can mess with his mind because he's had very specific training to prevent dream info extraction.
ARTHUR: Fischer’s had an extractor teach his mind to defend itself. His subconscious is militarized. It should’ve shown on the research—
Thirdly, the attack on his mind is quite ingenious. They co-opt his own defences (using the Mr. Charles technique) and use his own subconscious desire to make peace with his father against him.
FISCHER: The will means that Dad wanted me to be my own man, not live for him. (turns to Browning) And I’m going to, Uncle Peter.
Browning nods. Wipes the rain from his face. In the puddle beside them, the reflection is not Browning, but Eames.
Fourthly, this is a dream. Although we see it very clearly, the reality for Fischer is much more chaotic and unformed. By the time he wakes up (from sedation) he very rapidly returns to normal. He's evidently forgotten the dream and the events therein and seems pre-occupied, presumably still shaken up by the recent death of his father.
Arthur smiles broadly at Cobb. Cobb brushes past Fischer, who glances back at him as if thinking maybe he should know him, then moves on…