That question is the subject of this short Holden Karnofsky essay. Many people told Holden it is the best album ever, some citing its use of the recording studio, and he tried to work his way through that claim, basically remaining skeptical. Here are various responses to him. Here is a piece explaining the wonders of Pet Sounds, it is OK enough but not so insightful. I would stress the following points:
1. It is an album of sadness, loss, and infinite longing. Melancholy. Do I know of a sadder album? Listen to the lyrics. And yet it is all set amongst the sunshine and girls and southern California. As for the harmonies, they are continually building up expectation and never satisfying it. It is necessary for the album to end on the down note of “Caroline, No,” a song which itself just fades away and ends, merging into the “pet sounds” that give the album its name. I think of the combination of the sadness and the rising and swelling but never satisfied expectations as the key feature of Pet Sounds.
2. It is worth a listen-through following only the bass lines. You also will hear the huge influence on Paul McCartney.
2b. It is worth a listen-through following only the harmonies. The bells. The percussion. The woodwinds.
3. I don’t even think it is the best Beach Boys album. Or sort of it is. Overall I find the Smile period to be more profound, noting that this material ended up spread out over a number of separate albums. That said, every single composition on Pet Sounds is excellent.
4. “You Still Believe in Me” and “I Just Wasn’t Made for These Times” I both prefer to “God Only Knows,” which perhaps I have heard too many times.
5. Overall I find the secret to the Beach Boys (and some Beatles) listening to be their sound world. Interpret the Beach Boys through John Cage! Listen to a simple song such as “Vegetables,” but on a very good sound system or with head phones. Surrounded by silence. Or pick some of the other works from the Smile period, or even Wild Honey or the top cuts on Sunflower, such as tracks 7-10. Try to discern the sound of the air behind the music, the silences, and the tautness of the sounds that are sent your way. Internalize that understanding (if you are trying this for the Beatles, pick the noises at the end of “You Never Give Me Your Money.”) Carry that understanding of the sound world with you every time you hear a Beach Boys song. At first you will hear that sound world in the “pet sounds” at the end of the album, most of all the train, and then will you will hear it throughout the entire album.
Musical life will never be the same again.