Others in the GOP have raised concerns that their legislative hopes under unified Republican control could fade, given the confusion over Trump's priorities on issues such as tax reform and trade. "There are a lot of questions on the part of the people who took the President home after the dance," explains Republican Senator Pat Roberts of Kansas.
It says there are a lot of questions on the part of people who took the President home after the dance
I have a good guess what those sentences should mean but I have no idea who they supposed to be. I don't know much about American politics.
Source: http://time.com/4672974/donald-trump-white-house-chaos/
Best Answer
"On the part of" is an idiom that is synonymous with coming from. So, your sentence could be understood as "There are a lot of questions coming from the people who took the President home after the dance," (See http://idioms.thefreedictionary.com/on+the+part+of)
The "people who took the President home after the dance" is a colorful way of describing the president's biggest supporters, the ones who stayed enthusiastic about him after the election was over.