To “send them a message” is a call to action from a politician telling supporters to use their political capital to voice their opinion.
This can be performed in many ways. A protest can be sending a message, because it shows large support for or against an issue. The same goes for political donations. The most typical call for a message to be sent is through voting. A politician may call on the people to “send them a message” to America by voting for them, and showing that a majority of people support a certain policy.