The 23-year-old will join City on an initial two-year deal following the expiry of his deal at the County Ground at the end of next month.
The former Reading player has become City's fifth signing of a hectic May following the earlier additions of defender Matt Platt, forward Jake Young, keeper Harry Lewis and winger Jamie Walker.
East moved to Swindon last summer and made 23 appearances for the Robins, having previously made his debut for the Royals against Leeds United back in March 2019.
East said: “I think my time at Swindon sets me up well to play here. We did well last year, getting in the play-offs, and I want to go one better than that. The ambition of everyone at the club is clear.
“As a player, you do not want to go somewhere and be happy to sit mid-table. You want to be winning things, and I cannot wait to start playing in front of the fans.”
“I am absolutely delighted to be here. It is a massive club with a proud history, and I cannot wait to get going.
“When I met the manager, he sold it to me straight away. He wants to play a style of football that suits me. I like to have the ball on the floor and move it around.
City manager Mark Hughes continued: “Ryan is a player we are very excited about, and are really pleased to be able to bring to the club.
"His intelligence - both in and out of possession - and reading of the game shows maturity beyond his years. He has the ability to dictate play, and fits in well with the brand of football we want to implement.
“At 23, he is still young and at a great age to develop, and we have been impressed by what we have seen of him so far in his career."