Why radian is dimensionless? Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r

Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?

A quantity is dimensionless if it has same magnitudes in different units.
$$1 \text{rad}=\dfrac{1m}{1 m}=\dfrac{1 nm}{1 nm}=\dfrac{1\text{light year}}{1\text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1\ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
