It's difficult to store electricity. The peak hours of 4-7 are when most people are looking for power and when the demand may match or be more than the capacity of the grid to supply it. If there are limits on the amount of gas/oil/coal etc available to burn for more traditional power stations, and the wind doesn't blow, sun has gone down for the night and there is not a lot of water in the rivers where there is hydro, then there will be a need to cut power to some areas so that others still have it and there isn't a cascade failure of the entire grid.
The demand is far lower at 10pm-6am, so even if the wind and sun are lower, the traditional stations may be able to cope at that stage.
If the wind is blowing well and it is still sunny, there is more capacity available.
There is a hydro station in Ireland (Poulaphouca?) where the water from the upper lake runs through the turbines to generate electricity at peak demand times. The water runs into the lower lake from the turbines. But during the night, when demand on the system is low and the traditional stations need to be online anyway (and especially if it is a windy night and there's LOADS of power!), they can use that power to pump water back up to the upper lake to be available again tomorrow for extra peak supply. Which is great if the normal amounts of rain fall and we haven't had a drought emptying the lakes/rivers. (At least we're not as bad as Lake Mead/Hoover Dam hydro stations near Las Vegas in the USA!).
But it's hard to store power on a large scale otherwise. We don't have massive banks of batteries to recharge - but it is a reason why recharging electric cars overnight is a good thing to take that excess power and make it useful. There is research going into converting electricity into Hydrogen and storing that, and converting it back to electricity or to use it to fuel cars (whole different engine system!), when it is needed.
But the other thing that WE can do, as people living here, is to try and flatten the demand curve by reducing what we want at the peak and using it more spread out. So this is why people talk about running the washing machine and tumble dryer and dishwasher earlier in the day or later in the evening. Or using a slow cooker or having the oven on a timer to cook dinner earlier in the day before the peak, so it just needs a small heat through later. Or fill the oven one night (dinner for that night - meat, potatoes, veggies; as well as cooking other bits to just quickly reheat tomorrow or do some baking) and then not need the oven on the following day as that takes a lot of energy to heat. Don't turn all your devices on to recharge as soon as you all get in from work/school at 5pm but wait until after dinner and cleaning up when the overall demand on the grid is a bit lower. Lag your hot water cylinder and heat the water overnight to have your shower in the morning. Wait until there's a windy day to wash the sheets and towels as the wind will also give more power to the grid (and then hang them on the line to dry as well as the wind will get them soft and dry in no time without the tumble dryer 
).
There should still be plenty of power to do everything - it's just a case of thinking about when certain things NEED to be done (like needing to warm up from travelling home and feeding everyone at dinner time) and if you can organise your household to do certain other things at a slightly different time. So it's about rationing when you do things rather than rationing being able to do them at all.