When you act "in someone's name", it means that you act as if you were that person. For example, if Jack is not capable to act on his own behalf due to sickness or dementia there might be a person, Bill, who is assigned legal authority to act "in Jack's name". Bill would have the amazing power to "be" Jack. For example, he could buy a house using Jack's money and the house would belong to Jack, even though Jack had nothing to do, really, with the purchase. The phrase is used in many other contexts that have no relation to law. For example, religious people might pray "in the name" of a someone considered holy. That means, I think, that though they might not be worthy to talk directly to God, He might listen to their words if channeled through that holy person. (Sorry, hard to say this exactly, but something like that.) The phrase is also used in politics. If a country does something a citizen does not like, the citizen might say his government is not acting "in his name".

"their name" refers to "the people" in the previous sentence. Those who nominated the president acted representing (in the name of) everyone who elected them. "then" is just a synonym of "therefore", "because of that".

"in their name" = all the people's authority then = in that case