The Japanese manufacturer has not operated its own team since 2008 but returned to the series as a power unit supplier in 2015.
"As a team we understand how hard it has been for Honda Motor Company to reach the decision to step back from F1 at the end of the 2021 season", said Horner.
Honda has set itself a goal of becoming carbon neutral by 2050 and as a result plans significant investments in the areas of battery- and fuel cell-electric technology.
Red Bull-Honda took three victories and two pole positions last year and has added to its win tally this year to emerge as Mercedes' primary challenger, amid Ferrari's demise. "Also, I would like to express our appreciation to the FIA, Formula One and many people who were involved for all the support they have given to Honda since we chose to compete in the F1".
Honda has indicated that it communicated its thoughts to Red Bull in August, and made its final decision at the end of September - in other words some time after the Concorde was signed, and at a time when both teams still believed that their futures were linked to Honda.
The statement went on to confirm that Honda will continue to supply Power Units for Red Bull Racing and Scuderia Alpha Tauri for this season and into the next, with the brand new Power Unit already being worked on. However, both Mercedes and Ferrari are unlikely to strike a deal with Red Bull and Alpha Tauri, with Renault now being the most probable supplier for the two teams.
Pierre Gasly's victory for AlphaTauri at this year's Italian Grand Prix saw Honda become the only power unit manufacturer to win with two different teams since the start of Formula One's hybrid era in 2014.
Red Bull Racing team principal Christian Horner said: "As a team we understand how hard it has been for Honda Motor Company to reach the decision to step back from Formula One at the end of the 2021 season".
"For next year we will launch our new power unit so that together with Red Bull we are going to aim for the season championship".
"Their decision presents obvious challenges for us as a team but we have been here before and with our strength in depth we are well prepared and equipped to respond effectively, as we have proven in the past".
Honda's announcement comes as Red Bull Racing's annual account revealed the team spent UK£237.3 million (US$304.84 million) in finishing third in Formula One last season.
Of course the move means that Red Bull and AlphaTauri will be without an engine supplier for the 2022 season and the start of the new regulations.