With some verbs the distinction is easy to see, e.g.
Don't pick me. versus Don't pick on me."
The use of 'on' turns 'me' from a direct object into an indirect object.
'on' in this context means 'with respect to'
Don't pick me. means Don't choose me.
Don't pick on me. means Don't pick/choose (a fight) with respect to me.
You should cut your drinking means literally You should remove your drinking (from your life)
You should cut down your drinking means You should reduce the size/quantity of your drinking.
You should cut down on your drinking means You should make a reduction with respect to your drinking
I hope you can see that using 'on' allows us to remove the direct object from a sentence containing a verb that requires one (i.e. a transitive verb), and substitute an indirect object. We unconsciously accept that the direct object is there without needing to know precisely what it is. In my examples, I added 'fight' and 'reduction' but someone else might have picked slightly different words with a similar meaning.
I hope that helps.