BEIJING—Chinese leader Xi Jinping got out of quarantine and made his first public appearance since visiting Central Asia, trumpeting his governance record over the past decade as he prepares to extend his rule for a third term. Until Tuesday, Mr. Xi hadn’t appeared in public since his trip earlier this month to Central Asia , where he met with Russian President Vladimir Putin , because he was abiding by quarantine protocols that China imposes on people returning from overseas travel, according to people familiar with the matter.