Definition of Wavefront set is given as follows
Let $u$ be a distribution in $\mathbb{R}^{n}$. We say that $u$ is $C^{\infty}\left(\right.$ resp. near $\left(x_{0}, \xi_{0}\right)$ if there exist $\varphi \in C_{c}^{\infty}\left(\mathbb{R}^{n}\right)$ with $\varphi=1$ near $x_{0}$ and $\psi \in C^{\infty}\left(\mathbb{R}^{n} \backslash\{0\}\right)$ so that $\psi=1$ near $\xi_{0}$ and $\psi$ is homogeneous of degree 0, such that $$ \text { for any } \left.N \text { there is } C_{N}>0 \text { so that } \psi(\xi){(\varphi u)} ^{\hat{}} \xi\right) \leq C_{N}(1+|\xi|)^{-N} $$ The wave front set $W F(u)$ consists of those points $\left(x_{0}, \xi_{0}\right)$ where $u$ is not $C^{\infty}$.
I do not understand how decay property in $\xi$ leads to smoothness?
Any help or hint will be greatly appreciated.
It's a statement on localization of the Fourier transform of your localized distribution. Decay of the Fourier transform of a function/distribution is linked to smoothness of the function/distribution and vice-versa. You should contrast your definition with the Paley-Wiener theorem: https://en.wikipedia.org/wiki/Paley–Wiener_theorem. An alternative definition is $(x_0,\xi_0)\notin \text{WF}\ u$ if and only if there exists $P\in\Psi^0$ that is elliptic at $(x_0,\xi_0)$ such that $Pu\in C^\infty.$ Apply elliptic regularity to see the smoothness.
You can also see the smoothness from the fact that the projection of the wavefront set onto $x$ is exactly the singular support.
EDIT: Here is a set of notes written by Jared Wunsch that you might find useful: https://sites.math.northwestern.edu/~jwunsch/micronotes.pdf. It references both definitions.