How do I reference the input of an HTML control in codebehind?

You are not using a .NET control for your text area. Either add runat="server" to the HTML TextArea control or use a .NET control:

Try this:

<asp:TextBox id="TextArea1" TextMode="multiline" Columns="50" Rows="5" runat="server" />

Then reference it in your codebehind:

message.Body = TextArea1.Text;