If everyone goes around mowing lawns for each other, the economy is richer in lawn mowing at the expense of all the other things that would have been funded had everyone mowed their own lawns and purchased different services instead.
I am confused with this, if "everyone mowed their own lawns" then the net result will be exactly the same, everyone will be busy the same and not poorer, just without money movement.
This is not the same. If everyone wants mowed lawns, and everyone is busy working on that, there is no opportunity cost, everyone is working on their top priorities. The broken window fallacy is a fallacy because the headline gdp figure doesn't account for the destruction of the window which cancels out the benefit. In the grass mowing analogy nothing has been destroyed, useful and priority work has been done all around.
If everyone goes around mowing lawns for each other, the economy is richer in lawn mowing at the expense of all the other things that would have been funded had everyone mowed their own lawns and purchased different services instead.