It roughly means, "I start to..." but not just any word can fit in there. Think of it as putting on a pair of pants, except it's a word which relates to the action.
Eg. She began to cry - She got her weep on.
I start walking - I get my stroll on.
It's better to just listen to what is being said, and then you'll get a taste for what sounds "right". Here are a few examples sifted from the net:
I get my swag on. - I start to walk/behave with attitude (ie. swagger).
I get my grown man on - I do something which makes me seem grown-up, eg. how I dress.
It's time to get my preach on - it's time to start to lecturing people.