In the first chapter of The Stranger, Meursault shows no emotion at the news of his mother's death. He is also quite stoic or even casual during the vigil, commenting more on the sights and sounds (i.e. women coughing) than he comments on his own feelings on his mother's death. And when he does comment on these feelings, there is a lack of passion and no feeling of loss. When his mother's friends come to keep vigil with him that night, they seem emotionally affected. Meursault does not. In fact, they sit there in silence and Meursault feels that they are critical of him and his lack of grief.
I inclined to think that they were greeting me, after their fashion, but it had a queer effect, seeing all those old fellows grouped round the keeper, solemnly eying me and dandling their heads from side to side. For a moment I had an absurd impression that they had come to sit in judgment on me.
It is Meursault's cryptic and seemingly uncaring behavior, and others' reaction to him, that gives him this feeling of alienation. When Meursault doesn't react the way so called "normal" people react, those around him are baffled and this compounds his alienation.
After being jailed, Meursault has meetings with the magistrate. The magistrate says it is "unthinkable" that Meursault doesn't believe in God. During these initial meetings, Meursault feels alienated and misunderstood but as the magistrate adjusts to Meursault, he (Meursault) comes to feel "absurdly" like "one of the family.
Finally, Meursault accepts his fate and argues his way out of being sorry for himself because he believes it doesn't matter, in the grand scheme of things, when his death comes. He accepts that life is absurd and this justifies the way he lived his life. The last lines of the novel:
To feel it so like myself, indeed, so brotherly, made me realize that I’d been happy, and that I was happy still. For all to be accomplished, for me to feel less lonely, all that remained to hope was that on the day of my execution there should be a huge crowd of spectators and that they should greet me with howls of execration.
That is, if life is absurd, who's to say how it should be lived? In the end, he feels "so brotherly" and "so like himself" as if he is his only real brother, alienated from all others. He has even, at this end, embraced the alienation of his own execution. His last hope is that he will be greeted with "howls of execration."