I came across the phrase 'I got this' in an episode of 'How I Met Your Mother'. In the episode, Robin kept saying 'I got this' whenever something came up that needs dealing with. I guess it means 'Leave this to me. I can handle it'. Urban Dictionary says it's short for 'I got this under control.' I think the omission makes sense.
The thing I don't understand is why past tense is used here. When you use 'I got this', you are referring to something in the present, not in the past. Maybe you can get it under control, but you haven't gotten it under control yet. Why do you use past tense?
Here is an example from Urban Dictionary:
You're are driving a truck full of your stoner friends. The car in front of you slams on its breaks and all of your wasted friends are screaming that they are going to die.
You scream "I GOT THIS!!!" and then you stop your car just inches away from the car's bumper.
It is very clear that you had screamed 'I got this' before you actually got it under control.