Mr Cummings was speaking at an event marking the launch of a new book by Vote Leave supporter and businessman Stuart Wheeler.
When asked if he blamed MPs for the abuse they were getting, Mr Cummings said: "The MPs said we will have a referendum, we will respect the result and then they spent three years swerving all over the shop.
"It is not surprising some people are angry about it. I find it very odd that these characters are complaining that people are unhappy about their behaviour now and they also say they want a referendum. How does that compute for them?"
Speaking at the event, held at Carlton House Terrace in Central London on Thursday, Mr Cummings added: "To me it says that fundamentally a lot of people in Parliament are more out of touch with the country now, than they were in summer 2016."
Mr Cummings added: "If you are a bunch of politicians and say that we swear we are going to respect the result of a democratic vote, and then after you lose you say, we don't want to respect that vote, what do you expect to happen?"
When asked if the MPs have themselves to blame for the abuse, Mr Cummings said: "That's the way you're putting it. I am using my language."
When asked if the language used around Brexit was putting MPs and public figures at risk, Mr Cummings said: "People on all sides have said things that veered between unwise and very unpleasant, and sometimes criminal.
"That is true of people of the Leave side and that's true of people on the Remain side.
"People have been running around during the referendum campaign saying I was a Nazi, they run around for three years afterwards saying I am a criminal, and now a criminal Nazi.
"And there are also a bunch of people on the Leave side who have said terrible things about Remainers."
Mr Cummings added: "I don't think anyone is going to listen to reason because a lot of people (MPs) become really badly disconnected from what people in the real world, and England outside central London, thinks."