WASHINGTON—President Donald Trump’s campaign team was briefed before and after foreign-policy adviser Carter Page’s trip to Moscow last year, with Mr. Page saying upon his return that he had gotten “incredible insights and outreach” from senior members of Russian President Vladimir Putin’s administration, newly released testimony reveals.

Mr. Page, an unpaid adviser to the Trump campaign, acknowledged under questioning last Thursday in front of the House Permanent Select Intelligence Committee that he had met with Russian...