According to wikipedia
If $X = \mathbb{R}^{n}$ is the $n$-dimensional Euclidean space, and if $\gamma$: $[a,b]\rightarrow\mathbb{R}^{n}$ is an injective and continuously differentiable function, then the length of $\gamma$ is defined as the quantity
$$\text{Length}(\gamma) = \int_{a}^{b}|\gamma'(t)|dt \tag{1}\label{eq1}$$
The length of a curve is independent of the parametrization $\gamma$
In particular, the length $s$ of the graph of a continuously differentiable function $y = f(x)$ defined on a closed interval $[a,b]$ is
$$s = \int_{a}^{b}\sqrt{1 + [f'(x)]^{2}} dx \tag{2}\label{eq2}$$
My 2 questions are :
what is the meaning of the $df$ sign on top of the equal sign of the wikipedia equation (which I couldn't reproduce here)
what is the intuition behind the second equation (equation (2))
Df is the shorthand for "definition" here. You might have seen the sign $:=$ before.
The intuition behind the second equation is as follows. For small enough change $dx$, function $f$ changes by amount $df$. The length can then be approximated by Pythagora's theorem to be $\sqrt{dx^2+df^2}$. If you want the full length, sum over all the small changes to get $\int_a^b \sqrt{dx^2+df^2}$. Finally:
$$\int_a^b \sqrt{dx^2+df^2} = \int_a^b \sqrt{1+\left(\frac{df}{dx}\right)^2}\, dx = \int_a^b \sqrt{1+f'(x)^2}\,dx.$$
Of course, you can just use the definition of the length of the curve as given in the article if you choose to parametrize the graph as $\gamma(t) = (t,f(t))$ and so $\gamma'(t) = (1,f'(t))$.