What is the physical OR geometrical interpretation of the exterior derivative?

Question

The exterior derivative of a $C^{\infty} 0-$ forms on an open set $U$ of $\mathbb R^n$ look like the total derivative of $f$ in the calculus.

My doubt-

1. Why did an exterior derivative of $C^{\infty} k-$ forms on an open set $U$ of $\mathbb R^n$ defined like this?

2. What is the physical OR geometrical interpretation of the exterior derivative?