The World Cup winner saw a switch to Major League Soccer announced earlier this month.
United were prepared to let the 32-year-old go after seeing him become a peripheral figure under Jose Mourinho.
Schweinsteiger was frozen out of the picture completely at Old Trafford upon the Portuguese’s arrival, although he was later welcomed back into the fold.
Game time proved to be elusive, though, and the experienced midfielder is now ready to embark on a new challenge in America.
Robben, who previously worked with Schweinsteiger at Bayern Munich, believes his former team-mate has made the right call at this stage of his career.
He told Sky Germany: "That is his decision. He had a very difficult time in Manchester, it didn't work out like he wanted to.
"Now it's a new step, a new challenge, a new experience. So I can understand him. I hope he has a wonderful time."
Schweinsteiger moved to England in 2015 having spent his entire professional career up until that point at the Allianz Arena.
While things may not have gone to plan for him in the Premier League, a man with 121 international caps and 500 appearances for Bayern to his name is still held in the highest regard by Robben.
The Dutch winger added: "If you talk about Uli Hoeness, Philipp Lahm, Bastian Schweinsteiger - they are legends of FC Bayern."
Schweinsteiger officially completed his move to the Fire on Wednesday after seeing a visa issue resolved.