I’m no expert, but I think you’re mixing up jail and prison. Prison would require a judge, jury and trial. But a cop can unilaterally throw someone in jail temporarily until their first court appearance.

From the article:

They [the sherif and a deputy] told Patterson to turn around and put her hands behind her back. As three of her kids watched, Patterson was handcuffed. The sheriff took her purse and phone, put her in the cruiser, and hauled her off to jail.