I do not know if it's the simplest way, but just sending this information to another page (.php) can be as simple as that:
. index.html
<form action="saida.php" method="post">
<input type="text" name="nome"> // o name é usado para resgatar no PHP
<input type="text" name="email">
<textarea rows="4" cols="50" name="mensagem"></textarea>
<input type="submit" value="Enviar">
</form>
. saida.php
if (isset($_POST)) { // o isset verifica se o formulário foi enviado
$nome = $_POST['nome'];
$email = $_POST['email'];
$mensagem = $_POST['mensagem'];
echo "Nome: " . $nome . "<br>";
echo "Email: " . $email . "<br>";
echo "Mensagem: " . $mensagem . "<br>";
}
else {
echo "Falhou.";
}
This is a very simple example, without any kind of validation and sanitization of the data, and if you want to send the result to an email, file or database are already another five hundred.
To submit on the same page, see this question.