We are seeing at representations $sl_3(\mathbb C)$ here from Fulton and Harris. The space $\mathfrak h$ is collection of diagonal matrices and we have written $\mathfrak{sl_3}(\mathbb C) = \mathfrak h \oplus (\oplus \mathfrak g_{\alpha})$ which is obtained by adjoint action of $\mathfrak h$ suhc that for any $H \in \mathfrak h,$ $Y \in \mathfrak{g}_{\alpha}$ we have $[H,Y] = \alpha(H)Y$ where $\alpha$ is a linear functional on $\mathfrak h.$ Then we find that the eigen vectors are $3 \times 3$ matrices $E_{i,j}$ with $L_i$ as eigen value where $L_i\begin{pmatrix}a_1 &0 & 0\\ 0& a_2 & 0 \\ 0 &0 & a_3 \end{pmatrix} = a_i.$
Then the author draws the above picture that I do not understand the motive. How did they draw this?
The author claims that
The virtue of this decomposition and the corresponding picture is that we can read off from it pretty much the entire structure of the Lie algebra. Ofcourse, the action of $\mathfrak h$ on $\mathfrak g$ is clear from the picture: $\mathfrak h$ carries each of the subspaces $\mathfrak g_{\alpha}$ into itself, acting on each $\mathfrak g_{\alpha}$ by scalar multiplication by the linear functional represented by the corresponding dot.
How exactly they are claiming that one can know the structure of Lie algebra using the picture? Please provide some intuitions.
That is a diagram depicting the $6$ roots of $\mathfrak{sl}_3$. Those are the each of the $L_i - L_j$. Any semisimple Lie algebra is given by the sum of a Cartan subalgebra and its root spaces and moreover you know how the bracket works between any two root spaces ($[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}$) so this diagram conveys pretty much the whole structure of $\mathfrak{sl}_3$.
They have drawn those $6$ roots plus the $0$ weight on the backdrop of the weight lattice which is just the integer span of the weights $L_1, L_2$. All weights of finite dimensional representations live on the weight lattice (including the roots which are weights for the adjoint representation) so this is a natural thing to draw.