Why the long takes in Rope?

In 1962 François Truffaut interviewed Alfred Hitchcock rather extensively.
About 12 hours of audio has been broadcast on french radio. You can listen to the 25 parts here.

In part 15 Hitchcock talks about "Rope" (I transcribed it myself, so please excuse any errors):

I don't know why I really indulged in the stunt, I can only call it a stunt, in "Rope".

The play played in its own time. It was continuous action from the moment the curtain went up and when it finished. And I got the idea, how can I technically treat this in a similar thinking, that the technique should play without a break... the presentation of the film.

In other words, there is no break anywhere in the storytelling, It starts at at seven-thirty and finishes about nine-fifteen. And I got this crazy idea of saying, well, maybe if I could do it in one shot, the whole film.

When I look back of course it's quite nonsensical, really, because I was breaking all my own tradition of using film, the cutting of film, to tell story.*

The interview about "Rope" is about 30 mins long, so if you want to learn more about the film, have a listen.

The late film critic Roger Ebert wrote this about "Rope":

Alfred Hitchcock called “Rope” an “experiment that didn’t work out,” and he was happy to see it kept out of release for most of three decades.

To add to Christian Rau's answer, here is an outtake from Wikipedia:

When filming Rope (1948), Alfred Hitchcock intended for the film to have the effect of one long continuous take, but the cameras available could hold no more than 1000 feet of 35 mm film. As a result, each take used up to a whole roll of film and lasts up to 10 minutes. Many takes end with a dolly shot to a featureless surface (such as the back of a character's jacket), with the following take beginning at the same point by zooming out. The entire film consists of only 11 shots. (For a complete analysis of Hitchcock's hidden and conventional cuts in Rope, see David Bordwell's text "Poetics of Cinema", 2008).

Some will conjecture the reason you do not see much of this type of cinematography today is that the caliber of actor has gone down. Directors rely on action and imagery to tell the story rather than relying on the actor. The actor has become the "pretty face" to look at and not the means by which we learn what is going on. Hitchcock was renowned for his "long shot" style of story telling. And one of the reasons his movies still stands today as some of the best ever produced.

I think that the long takes, or just a single take as it should look for the audience, fit perfectly to the fact that the whole movie is set at a single small place. Those two aspects contribute to the impression that we are actually watching a play instead of a movie (as also stated in this answer to a different question).

To me this had two at first paradoxical effects. On the one hand it makes the time feel smooth yet short and thus emphasizes the real-time aspect of the action and in this way it contributes to the intensity of the whole situation. But on the other hand this smoothness also doesn't make it feel hectic or too fast, which emphasizes the calmness and intelligence of Brandon's actions and the mental games he's playing with his guests.

So this real-time one-take shot does neither delay the tension, nor does it artificially compress the action. It just shows it as intense as it really would have been.

(Though, these are just some small speculative points for now.)