The principle that no person can purchase or sell a firearm without first receiving government permission transforms the Second Amendment from a "right" that should be protected by the government into a privilege granted by the government.
In addition to constitutional concerns, even if every private transfer of firearms were regulated by the federal government, it would not be an effective crime fighting tool. Typically, criminal shooters steal firearms (Adam Lanza), pass a background check (James Holmes and Jared Loughner) or receive their firearms through straw purchasers (which is already illegal).
The letter includes a list of gun rights groups who join the representatives in opposing gun legislation currently being discussed in the Senate: the Gun Owners of America, the NRA, the National Association for Gun Rights, and the Citizens Committee for the Right to Keep and Bear Arms, among others.
The letter close with the forty-five representatives asking "that no gun control legislation be brought to the floor of the House unless it has the support of a majority of [their] caucus."