First, in the Citizens United campaign finance decision the Court determined that there was insufficient evidence that the restriction on campaign expenditures was justified by the government interest in preventing corruption or the appearance of corruption. In the Senators' view, the evidence gathered by Congress--as well as Congress's determination based on that evidence, a determination reflected in the legislation itself--provided adequate support for the conclusion that unrestricted expenditures leads to at least the appearance of corruption, if not actual corruption. But the Court found to the contrary. And the Senators claim that the Court failed to defer appropriately to Congress's superior ability to gather and assess evidence as well as to Congress's role as the policymaker.
The second line of attack involves statutory interpretation. Some of the Justices make their decisions about a statute's meaning by looking only at the statutory text and structure. Others are willing--at least when the text is ambiguous--to look at Congress's deliberations as reflected in the committee reports and floor debates. Ignoring this material, the Senators say, means that Justices are closing their eyes to Congress's will (the reason for those Justices' position is that these legislative materials do not reflect the views of all of the members of Congress who voted for the legislation--indeed, many of them probably did not even read them--and therefore are not reliable; also, they fear that these materials can be manipulated by congressional staff and interest groups).
Many members of Congress believe that the Court's lack of any Justice with experience running for elective office and serving in a legislature (whether a state legislature or Congress) leaves the Court with no real understanding of the political and legislative processes and even perhaps with an unjustified negative view of the way the political branches operate. It is worth noting that this state of affairs is unique in our history. For most of the Court's history it had a number of sitting Justices who had served in Congress or state legislatures. (For example, the Court that decided Brown v. Board of Education included a former Governor and former Senator.) Most recently, Justice O'Connor provided that link, having served in Arizona's legislature before becoming a judge.