I strongly recommend you to go to the Codelab for App Maker. The section Building a form to send an email describes the whole process.
The steps to highlight are:
Step 11 - Set the onClick property of the button as a custom action with the code:
var widgets = widget.parent.descendants;
var to = widgets.To.value;
var subject = widgets.Subject.value;
var msg = widgets.Msg.value;
widgets.EmailStatus.text = 'Sending email...';
SendEmail(to, subject, msg)
Step 13 - Add the following ClientScript code:
function clearEmailForm(){
var formWidgets = app.pages.Basic.descendants;
formWidgets.EmailStatus.text = "";
formWidgets.Msg.value = "";
formWidgets.To.value = "";
formWidgets.Subject.value = "";
}
function SendEmail(To, Subject, Msg){
var status = app.pages.Basic.descendants.EmailStatus;
google.script.run.withSuccessHandler(function(result) {
status.text = 'Email sent...';
clearEmailForm();
})
.SendEmail(To, Subject, Msg);
}
Step 14 - Now add the corresponding code to the ServerScript.
function SendEmail(to, subject, msg){
MailApp.sendEmail(to, subject , msg);
}