You've got several options that you could use. The first would be to use PHP to handle your form and append a text file that you have in a directory that is secured using the .htaccess file. Most hosting companies (at least the decent ones) have a way of creating password protected directories so you don't even have to mess with the .htaccess file.
The second option would be to store them in a database which for email addresses would be fine, at least I would think so. Without knowing the username and password to the MySQL database a bot or random user won't be able to connect to the database to submit queries on the data.
The third option would be to use the script you have to encrypt the email address using javascript.
I think simply handling it, and creating the email with PHP would be the easiest way to do it though. From there you can generate your email with the contents being in a comma delimited format. From there you can copy and paste it into a spreadsheet or database.
Try this link for a simple how-to
Form Handling with PHP
For the mail handler, if you wanted name, email address format the body part like this, and drop the fields you don't want.
Code:
$body = “$name, $email”;