The power grid is capcitized to handle peak demand - say, a very hot and humid Tuesday in the midwest. Utilities max their output and buy their power from others in areas less affected to keep all the lights on. Relatively few people are charging EVs mid-day, so it is not a significant drain on the resource.
For the rest of the time, the grid is way under its max capacity and there is a surplus... your basic cold rainy Sunday evening in the midwest. Plugging in at night just like your phone is how the vast majority of users will charge - filling in the curve if you will.
EVs have the opportunity in wider distribution to be a massive distributed storage resource. Assuming they are connected to the grid, it is entirely possible (GM is working on it now along with many others) to flow the stored energy BACK to the grid as needed to knock some money off your bill.