This is a political question, but we must look at the timeline.
26 or 27 August 2020: https://www.jsonline.com/story/news/crime/2020/08/26/illinois-teen-charged-homicide-kenosha-protest-case/5633334002/
"Court records show Kyle Rittenhouse, 17, of Antioch, Illinois, was charged in Lake County, Illinois, as a fugitive from justice. That document, reviewed by the Milwaukee Journal Sentinel, said he faces a first-degree intentional homicide charge in Kenosha County."
"Since Rittenhouse was taken into custody in Illinois, Wisconsin must file documents to extradite him to the state to face homicide charges in Kenosha County."
In other words, it was absolutely relevant during the arrest that Rittenhouse lived and was arrested in Illinois, but was accused of homicide in Wisconsin.
He therefore 'crossed state lines' to return to his home, as many millions of Americans do every day, so from the beginning the 'state lines' was relevant.
We see this article of 27 August 2020, which employs strident rhetoric:
"Rittenhouse traveled to the protests in illegal possession of an assault weapon to act as a vigilante. He placed himself at the center of the violence, then escalated it by shooting a man in the head. Minutes later, he shot two more people who appeared to be trying to disarm him, then fled across state lines. "
Much of that is not true (the gun is not classed as an assault weapon under Wisconsin law), but the 'state lines' issue arises because Rittenhouse crossed them, for simple reasons of geography.
L. Lin Wood, who sought to associate himself with the case, said https://www.jsonline.com/story/news/crime/2020/08/26/wisconsin-open-carry-law-kyle-rittenhouse-legally-have-gun-kenosha-protest-shooting-17-year-old/3444231001/
"Kyle did not carry a gun across state line"
So the 'state lines' were relevant to the process of arrest, and the confusion between the fact of Rittenhouse himself crossing state lines, and the gun, which did not, caused people to falsely claim that the gun had been brought across state lines.