'We cheer each other on before I depart.' - What does 'on' mean?

I’m not a native-speaker; sometimes prepositions (or adverbs) are tricky for non-native speakers.

‘We cheer each other on before I depart.’

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

(look at 2:10)

In this sentence, I don’t know what does ‘on’ mean? I understand what ‘on’ means in these sentences:

There’s a mark on your skirt.
Put your coat on.

But when we use ‘We cheer each other on’… What nuance of meaning of the word ‘on’ is here?

Is it possible to omit ‘on’?

‘We cheer each other before I depart.’ (Is this sentence correct?)

Would it still be the same meaning? Why does first sentence have ‘on’? And could you tell me some example sentences with this meaning of ‘on’ in them?