A few years after Carleson's proof Fefferman came up with a shorter proof of the $L^2$ and $L^p$ results. Later, in the context of their work in multilinear harmonic analysis Lacey and Thiele came up with a quite short and easy to understand proof for the $L^2$ theorem which is to some extent a descendant of Fefferman's proof. It's only 10 pages long and can be found in
Lacey, Michael; Thiele, Christoph (2000), "A proof of boundedness of the Carleson operator", Mathematical Research Letters 7 (4): 361–370.
(By the way, this paper has an amusing Mathscinet review which begins with "This is one of the greatest papers written in Fourier analysis.")
They also wrote an expository article describing this and a number of related results:
Lacey, Michael T. (2004), "Carleson's theorem: proof, complements, variations", Publicacions Matemàtiques 48 (2): 251–307
They also put an expanded version of this last paper on the arxiv http://arxiv.org/abs/math/0307008v4