How can I limit the submissions to the user who submitted the form?

I noticed that when a confirmation email is sent to the person who submitted the form, at the bottom of the email there is a link to edit the submission and a link to view all submissions.  I would just like the person who submitted the form to see only his submissions, not all of the other user submission from different users.

Is that possible?

The EDIT and VIEW submission links are only shown on Email Notifications (these are the emails you get as the form owner).

Related guide: Setting-Up-Email-Notifications 

Autoresponders on the other hand (the ones your users/respondents get) don't have these links by default.

Related guide: Creating-a-Form-Autoresponder 

No one else can view someone else's submission data except you as the account owner. Unless you purposely shared your credentials with someone, added a subuser, or shared a generated submission report, then there's nothing to worry about.

In case we misunderstood your question, just get back to us with a bit more info so we can better assist you.