That's not really what happened. Friction between Mercedes and McLaren existed on several levels.
First there was the Mercedes SLR road car project that caused a lot of frustration on both sides.
Mercedes had a 40% stake in McLaren, and wanted to buy the F1 team to rebrand it as a Mercedes works team. Ron Dennis refused.
McLaren had road car ambitions and developed the MP4-12C. Mercedes did not like that, as they viewed it as direct competition for their high-end sports cars.
When Honda withdrew from F1 at the end of 2008, their team principal Ross Brawn took over the team, minus the engine. He asked Mercedes for theirs, but McLaren had a veto. Their then-team principal Martin Whitmarsh agreed to let BrawnGP use the Merc engines for 2009, and Brawn bagged the double.
Mercedes had wanted to enter F1 as a works team anyway (originally by buying McLaren), and they scooped up BrawnGP at the end of 2009, gradually selling off their 40% in McLaren over the next four years.
And finally, when it became clear to Mercedes that the team they bought wasn't as good as their 2009 performance had suggested, they methodically started luring away technical staff from McLaren.