The First Amendment does not apply to corporations because the Constitution was established for "We the People" and was set up to protect individual, rather than corporate, liberties.
The BCRA leaves corporations other ways to speak and to spend money on elections. The law allows corporations and unions to form Political Action Committees and to fund advertisements through the PAC.
The Supreme Court has ruled on these issues before in Austin v. Michigan Chamber of Commerce and in McConnell v. FEC, which upheld the BCRA's bans. The Court should not completely change the law, which has clear public support.
Corruption is not limited to bribes and direct transactions. By being allowed to spend unlimited sums of money in support of a candidate, corporations and unions will have a certain amount of access to, if not power over, that candidate.
Corporations can accumulate so much money that they could overwhelm the conversation and drown out the speech of less wealthy individuals in an election.