To "obsess over" something means to spend too much time worrying about details.
For example, if someone spends two hours every morning to get ready to go out, you could say that they are "obsessing over" their appearance.
My brother obsesses over how he looks way more than I do.
This phrase sounds negative. To describe someone who pays attention to details in a positive way, you can say that they are "meticulous":
He's always been really meticulous about keeping his apartment clean.