Daisy and Tom invite Nick to dinner at their house because aside from Tom and Nick going to college together, Daisy is Nick's cousin twice removed. There may also be a hint that Daisy wanted Nick to meet Jordan Baker and maybe go out with her, though this is not fully known
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Nick agrees to arrange the meeting because he empathizes with Gatsby's love for Daisy and believes that they deserve a chance to rekindle their relationship. Additionally, Nick sees the potential for Daisy to find happiness with Gatsby, as he knows that Daisy has been unhappy in her marriage to Tom.
Nick is a tolerant open minded character who does not like to judge anyone. He has learned a lot about his cousin Daisy's marriage and a little bit about Gatsby. He agrees the arrange the meeting simply because he wants to help both his cousin and Gatsby.
I think it is because Nick admire's Gatsby romanticism. Nick, being a realist, wants to be as daring as Gatsby, especially of how Gatsby strives for Daisy.