In many English words "th" stands for a single sound in a word. You could consider it a single letter. There are two possible pronunciations for this 'letter'. One has your voice behind it (voiced) and the other doesn't (unvoiced). The final place 'e' in breathe functions the same way as in these sample word pairs:
pat --> pate
sit --> site
dot --> dote
In addition, it changes the 'th' sound from the unvoiced to the voiced:
It's difficult to transliterate the sounds of "breathe" and "breath", but if you have speakers to your computer, you might try looking up "breathe" and "breath" in the Wordreference dictionary and clicking on the icon that looks like a speaker to hear the pronunciation.