I have recently learnt about tempered distributions, and how one can define the Fourier transform of a tempered distribution $v$ to be $\hat v$ so that $$\langle\hat v,\varphi\rangle=\langle v,\hat \varphi\rangle.$$ for all Schwartz function $\varphi$. Here, let's say I take the convention that the Fourier transform is $$\hat\varphi(\xi)=\frac{1}{\sqrt{2\pi}}\int_\mathbb R e^{-i x\xi}\varphi(x) \, dx.$$ In particular, one can show that the Fourier transform of $1$ is the Dirac delta $\delta_0$. Here, $1$ is in the sense of tempered distribution, defined by $$\langle 1,\varphi\rangle=\int\varphi(x)\,dx.$$ No problem there.
However, I have come across a confusing usage of this fact: I was reading a proof of some estimate, the author claimed that ($g$ is just some function of $x$)
$$\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}\int_\mathbb{R} e^{-ixt}g(x)\,dt\,dx=\int_\mathbb{R}\delta(x)g(x)dx.$$ It looks like the author took the Fourier transform of $1$ "at the point $x$". But this is confusing to me because I don't understand what it means to take Fourier transform "at a point". I also don't understand what $\delta(x)$ means, and what it means to "integrate" it.
If the context is helpful, I am reading this set of seminar notes. https://www.math.ucla.edu/~visan/Oberwolfach2012.pdf Namely, page 11 Lemma 2.11 (Local Smoothing). I simplified the notation above the make the question less complicated.