Did you try clicking "Submit" yourself in the PDF they sent back to you?
It sounds like the person who filled out the PDF did not use "Adobe Reader" or "Adobe Acrobat". Reader or Acrobat are required to fill out FormsCentral forms as they have features not supported in other PDF viewing applications.
The best way to get that data into FormsCentral is to be able to submit the PDF. It may not work from that PDF if it was filled in another PDF viewer.
If it doesn't work you can ask the filler to fill out a new clean version of the PDF in Adobe Reader and they will be able to Submit. Otherwise you would have to add a new row to the Response Table (View Responses tab) and manually enter the data from the PDF (which you can do).
For some reasson it didn't have the Submit button any more.
If you open the file in Adobe Reader or Acrobat and use "Crtl+D" (Cmd+D on Mac) to bring up the Document Properties box there are two fields on the "Description" tab, both of those fields (Application, and PDF Producer) should be Adobe FormsCentral (one would say Adobe Acrobat if Acrobat was used to create the PDF), if either of those fields say anything other than Adobe FormsCentral or Adobe Acrobat the form filler used something else to fill out and save the PDF and that application broke it which it sounds like somehow even removed the Submit button (not sure if I've seen that one before).
If you can I would request the user fill it out again in Adobe Reader - send them a new clean PDF from FormsCentral so it hasn't already been damaged.