I go to the binThe singer is explaining that, having failed to do the laundry, and having no clean clothes he has to go to his dirty clothes and do the best he can with them. But because of the meter of the song, my mind keeps hearing it as if he is describing a series of thee actions: 1) he goes to the bin; 2) he throws the laundry in; and 3) he picks out his cleanest shirt. That's not what Lowe was trying to convey in the song, but I can't help hearing it that way.
I throw the laundry in
And pick out my cleanest shirt