They can't search the car without a warrant. They can't get a warrant without probable cause. They didn't impound his car until after his arrest.
Not many facts are being released yet, but CNN said they found him through genetic DNA. It was already stated his DNA was all over the crime scene. So, if they were able to match his to DNA from a relative in one of the ancestry databases, they could narrow down potential suspects. Then, they identify a relative who matches the profile, had means (was near the murder scene), and conclude he also has the kind of car placed at the scene. Then they track him to get his actual DNA to confirm a match. It's plausible.