I need to view the saved WiFi passwords on my non-rooted Android phone I have an LG Optimus 7 (not-rooted), and all my WiFi passwords are there. I want to view the saved WiFi passwords to transfer thm to my new Samsung Galaxy S3, but I don't know how.

I tried to search on the Internet and Google play for an application that would allow me to view this information, but I can't find one. Is there any way for me to accomplish this?
This is not directly possible (as noted in the comment below your question).

You can let Google save them to their cloud:

Go to: Settings->Privacy and enable those two options:  


Back up my data (Back up application data, Wi-Fi passwords, [...]) 
Automatic restore


Also, say yes when you're prompted if you want to Restore data during the first time set up of your new SGS3. If you've done the first time setup already withouth that option enabled, then maybe the Automatic Restore settings does it afterwards (not tried yet, though).