Alvarez falsely claimed in a public meeting to have received the Congressional Medal of Honor. This wasn't a misunderstanding, it was an out-and-out lie. As a result, he was convicted under the Stolen Valor Act, which makes it a crime to lie about receiving military awards. He appealed his conviction, challenging the constitutionality of the act.
The US Supreme Court found the act unconstitutional and set aside his conviction. As grounds for this finding, the court observed that content-based restrictions on speech are presumptively unreasonable. Although there are exceptions to this rule (in specific cases like perjury, fraud, fighting words, lies to government officials related to official investigations, falsely claiming to represent the government.... the list goes on), the mere falsity of speech is not an exception. Generally, there has to be a showing of a specific harm resulting from the false speech. Further, the government has to take the least intrusive means available to proscribe even unprotected speech. The Court suggests that in this case that could mean rewriting the act to include an element that the speech cause some harm to a specific person, or creating a searchable on-line database of legitimate recipients of the medal (private individuals have already done this). That way, public ridicule and rejection of false claimants should take of the rest. The remedy for false speech is true speech.
In any event, since there are less intrusive means available to sanction lies like Alvarez's, and since his attempts to bolster his own reputation don't meet any recognized exception to the First Amendment, his speech (while detestable) is protected.