Suppose $\phi_t$ is a (locally defined) One-parameter group of diffeomorphisms defined by vector field $X$. Then there is a naturally defined Lie derivative

$\mathcal{L}_X \alpha = \frac{\partial}{\partial t} \phi_t^* \alpha |_{t=0}$

of a p-form $\alpha$ by $X$. It is again a $p$-form.

See here

The Lie derivative for Vector fields is the Lie bracket.

Cartan's formula

A formula for the Lie derivative of a k-form using the Exterior derivative and Interior product

Intuition

As explained in Lie bracket, "So it says in which direction would I need to change the geometric thing $\alpha$ at each point, so that the flow $\phi_t$ induced by $X$ takes it to the original $\alpha$" (but this is not totally correct as can't generally define pushworfard, only Pullback)