(And this applies to any electoral reform that there is resistance to by incumbents, not just approval voting, which I don't particular prefer.)
In states with citizen initiative processes that bypass incumbent office holders, by that method. Elsewhere, by building support in the electorate and then running candidates on a platform of doing that, and keeping organized in the electorate to hold those candidates accountable.
(And this applies to any electoral reform that there is resistance to by incumbents, not just approval voting, which I don't particular prefer.)
In states with citizen initiative processes that bypass incumbent office holders, by that method. Elsewhere, by building support in the electorate and then running candidates on a platform of doing that, and keeping organized in the electorate to hold those candidates accountable.