Learn English – How to appropriately use the word “baby” as in the pattern ” ____ baby ___”; for example, “run baby run”

Flat out is an expression used to describe one of three things, depending on the context:

1. Doing something as hard or as quickly as you can: e.g. I worked flat out to finish the assignment.
2. Doing something without hesitating or any second thought: I flat-out disagree with your view.
3. Lying down completely stretched out: After winning the race, I was exhausted and collapsed flat-out on the ground.

In this case, it's the second or the first use, depending on the context, though in either case he's making a statement about his physical condition. The speaker is saying one of two things:

1. He can run as quickly as he can for half a mile before his hands start shaking
2. The speaker can, without a moment's hesitation, run half a mile, without a warm-up, before his hands start shaking.

This makes me assume that saying "My English isn't the best around" is probably wrong, but I'd like to know why exactly.

There is nothing wrong with the phrase. The reason you will only hear non-native speakers using it is because native speakers nearly always consider their English to be excellent in comparison to their peers.