CiviVolunteer Interest form actions
It adds them to CiviCRM and then you can search for them. You likely need to add a group or tag (I like a group, as I use it for mailing lists) to the settings for the profile that you are using. Then anytime someone fills out that form, they are added to your Volunteer group. Then inside CiviCRM you can easily run a search to pull the entire list of everyone who has signed up on that form.

And by using a group, you can then send them emails with volunteer opportunities - just make sure you select that it's for mailings when you create the group.