"How did all this indiscriminate drone warfare unauthorized by Congress get started?"

With the 2002 AUMF that gave presidents the authority to conduct strikes without congress authorizing them first. This was a--you guessed it!--Bush Jr policy proposal that became law in 2002 and has authorized US presidents to use military force (what AUMF stands for) without consulting congress if the targets of strikes fall under post-9/11 terrorism definitions. You'll recall that Obama once (controversially) killed an AmCit in Yemen under this statute.