The definitions of subdifferential and scaled subdifferential are (screenshot)
- Subdifferential: The subdifferential of $f$ at $x$ is the set of vectors $\partial f(\boldsymbol{x})=\left\{\boldsymbol{u} \in \mathbb{R}^{n}: f(\boldsymbol{x}+\boldsymbol{d}) \geq f(\boldsymbol{x})+\langle\boldsymbol{u}, \boldsymbol{d}\rangle\right.$ for all $\left.\boldsymbol{d} \in \mathbb{R}^{n}\right\} .$ For any number $\kappa \geq 0$, we denote the scaled (by $\kappa$ ) subdifferential as $\kappa \cdot \partial f(\boldsymbol{x})=\{\kappa \boldsymbol{u}: \boldsymbol{u} \in \partial f(\boldsymbol{x})\}$.
And there is the illustrations of the subdifferential, scaled subdifferential:
I don't know the meaning of the blues line, and what does the subdifferential set look like at the point $x = (1, 0)$?
The red surface is $z=\|x\|_1$. The green dot is $(x,y,z)=(1,0,1)$. The blue plane is $z=x$ which is one of many tangent planes to the graph at $(x,y)=(1,0)$, corresponding to the point $(x,y)=(1,0)$ in your diagram, which is in the middle of the left blue line.
As you can see, you can wobble the tangent plane in the $y$ direction to get other tangent planes $z=x+sy$ for $s\in[-1,1]$. This visually shows that $\partial f(1,0) = \{ (1,s) : s\in[-1,1]\}$ exactly as in your diagram.
As for the scaled subdifferential, it is a special case of scaling a set $kA=\{ka:a\in A\}$. For each point in $\partial f(1,0)$, draw the line from the origin to that point; if you keep going for a total of 2 times the distance, you end up precisely with the corresponding point in $2\partial f(1,0)$: