Why can't I hear sound except through headphones?

It's probably a faulty loud speaker.

I just fixed one last week by replacing it after testing the original one on there with a digital multimeter for continuity (but it was short circuited / open line meaning it would no longer work.)

The problem with just replacing it is that the screen would need to be removed first in order to get the loudspeaker / ringer speaker replaced. This requires very careful care while lifting up the LCD (which is held on by a lot of adhesive) and may end up resulting in a cracked LCD (no display).

If you are willing to take the risk of cracking the LCD and buy the replacement loudspeaker follow this guide to step 27 (You will need to pry out the loudspeaker as it has a adhesive under the speaker to hold it in):

Samsung Galaxy S5 Motherboard Replacement

You can buy the replacement part from here (Used is good as well, but may be slightly quieter than the original one due to being worn out):

[linked product missing or disabled: IF236-010-2]

You will also want a replacement screen adhesive as re-using the screen adhesive is probably not a good idea as it gets weakened every time the screen is lifted up (Make sure to clean screen and frame side before install):

http://www.ebay.com.au/itm/Digitizer-LCD...

Guide to install the screen adhesive:

https://www.youtube.com/watch?v=ApgB3ISm...