The Solidarity Service is an orgy. Its purpose is sexual release. The sexual release occurs in a context that binds people more tightly to the society as a whole rather than in an exclusive sexual relationship with one person. The orgy helps express a core ideal in the World State that "every one belongs to every one else."
The Solidarity Service is a parody of the Christian eucharist or communion service. In that service, people experience a oneness with Christ as they symbolically drink his blood and eat his body. As befits the emphasis on free sexuality in the World State, the "communion" has been changed to a group orgy. Because of censorship laws at the time, Huxley couldn't say explicitly that the group was having sex, but the intent is made clear in lines such as:
Boys at One with girls at peace;
Orgy-porgy gives release.
Influenced by drugs, suggestion, and throbbing music, most of the group has an ecstatic experience of the "Greater Being" in tandem with sex, offering them a pseudo-religious experience. Huxley parodies the Holy Spirit in a racist way when we writes:
in the red twilight it was as though some enormous negro dove were hovering benevolently over the now prone or supine dancers.
Bernard feels more alone than ever after the service because it makes him more acutely aware of how different he is from other people in his society. It has the opposite of the expected effect on him, and it leaves him "unreplenished," most likely because he realizes how shallow and superficial it is. He is uncomfortable because he has been conditioned all his life to want to be like everyone, but he is not like the others. He is more "hopelessly himself":
He had emerged from that crimson twilight into the common electric glare with a self-consciousness intensified to the pitch of agony.