As far as I understand, if an apostrophe is in place of some letter, this letter isn't pronounced. For example:
is not ['iz 'na:t] → isn't ['iznt]
about [ə'baut] → 'bout ['baut]
because [bɪˈkɑ:z] → 'cause [ˈkɑ:z]
But what does an apostrophe change, appearing instead of "g" in the ending "ing"?:
-ing [iŋ] → -in' [?]
doing → doin'
Maybe [iŋ] → [in], i.e. [ŋ] becomes [n]?