Tom Buchanan and his two upper-class friends, the Sloanes, are out horseback riding. They stop at Gatsby's merely because they are thirsty and want a drink of water. Tom wants to use Gatsby, as he does people not of his social class, for his own convenience. Gatsby mistakes this stopping in for a social call and treats his visitors as guests.
Nick happens to be there and registers embarrassment when Gatsby mistakes the slightly drunken Mrs. Sloane's dinner invitation as sincere. This shows that Gatsby buys into the American dream that anyone who makes enough money, as he has done, can enter the highest classes. Tom, of course, never for an instant sees Gatsby as anything more than a lower-class grifter. He jeers at him to Nick for taking the invitation seriously, and the three equestrians ride off without him.
It's significant that this scene takes place after Gatsby and Daisy have reunited. Gatsby has always seen in Daisy the confident girl and woman who grew up in secure and affluent financial conditions. Part of what attracts him to her is her casual attitude to wealth—she has never had to worry for one minute about money. Being back with her represents to him the acceptance he has always wanted into the upper classes. Because she is having an affair with him, he feels more confident than ever that he is accepted in her class. Although he usually has sensitive social skills, his love of Daisy blinds him to reality. He feels a closeness and affinity to Tom that Tom does not to him. It doesn't occur to Gatsby that Daisy may just be using him.
The scene reinforces that no matter what Gatsby does, he will never be part of Tom and Daisy's world.
During Gatsby's brief interaction with the Sloanes, he demonstrates his inability to read social cues and take a hint. When Mrs. Sloane casually asks him to attend dinner at their house, Gatsby enthusiastically accepts the invitation without realizing that she was just being friendly. Unlike Nick, Tom, or the Sloanes, Gatsby does not hail from a wealthy family and is unfamiliar with the social cues of the elite. Gatsby misinterprets Mrs. Sloane's invitation and foolishly believes that she is being sincere. In contrast, Nick understands that she is being decorous, and he politely declines the offer.
Gatsby's enthusiastic response also reveals his desire to meet up with Daisy and gain entrance into her social circle. Since Gatsby moved to the West Egg, he has been trying to reconnect with Daisy and prove that he belongs in her social class. Gatsby is willing to do anything to impress Daisy, which includes attending random dinners with other wealthy strangers. When Gatsby enters the house to get ready, Tom quickly criticizes him for his lack of social grace and is appalled that Gatsby accepted Mrs. Sloane's insincere invitation. Nick responds by coming to Gatsby's defense and mentions that Mrs. Sloane did invite him. Nick is subtly calling out Mrs. Sloane for being insincere, and the group leaves before Gatsby returns.
In Chapter 6, Tom Buchanan, Mr. Sloane, and a young woman arrive at Gatsby's home on horseback and briefly stop at his mansion for something to drink during their ride. After Gatsby invites Tom to dinner, the woman cordially invites Nick and Gatsby over to her home for supper. However, Mr. Sloane is a haughty man who does not approve of having Gatsby over for dinner and Jay does not take the hint. When Gatsby goes inside his home to get ready, Tom expresses to Nick that he cannot believe that Gatsby is actually coming to dinner while Mr. Sloane and the woman begin to argue. Jay Gatsby's response to the woman's invitation suggests that he is socially naive and innocent. He believed that her invitation was genuine, which is why he was preparing to go. He does not come from an aristocratic background and does not understand the social formalities of the upper class. Gatsby's motivation to attend the dinner may also have had something to do with seeing Daisy. Gatsby may have hoped to see Daisy at dinner since Tom Buchanan was obviously attending.
In our society we have people who suffer from mental or physical disabilities. This scene demonstrates another disability at work, social disability. Gatsby is not a good reader of people or their intentions. He doesn't get the hint... several hints, and it shows.
I think Gatsby is indeed paralyzed by his love for Gatsby and a chance to see her would indeed be great, but he may also want to be socially accepted. That would be an additional feature to have in order to demonstrate worthiness to Daisy. Here, he is awkward in and among a group of people. His parties aren't awkward... but perhaps that's because he doesn't engage with the people.
I think that this scene is meant in part to show that Gatsby is not very socially sensitive. Tom points out, once Gatsby is not in the scene, that the invitation was not really sincere. However, Gatsby does not seem to realize this.
However, I think the scene has more to do with Gatsby's obsession with Daisy. Gatsby has been hoping, of course, to find a way to casually meet up with Daisy again. He has been hoping to show her that he has become rich and worthy of her. Therefore, he will do anything (even if it is expensive or socially awkward) to try to meet her again.