Burnley were relegated from the Premier League in the first term after they were promoted.
Kompany could not keep them in the top flight, but has joined Bayern Munich in Germany.
“It starts with a message of thanks, of course. I was very lucky; I was surrounded by hard-working people, players who gave everything,” he said on Instagram.
“And I felt privileged and honoured to be welcomed, and to be part of such a good environment – so for that, thank you.
“The best way to describe how I lived it is that I think I gave everything back.
“(The) best memory I had was beating Blackburn at home 3-0 and then going and beating them at their home and winning the league there – that was good. I loved every moment at the club – it’s been a very important part of my life.”
