I've seen more than one time the use of the following notation: $(\vec{v}\cdot\vec{\nabla})\vec{v}$ to say $L_v(\vec{v})$ where the matrix $L_v$ is defined as follows: $(L_v)_{ij} := \partial_j v_i$.
For example this happens in the usual law of conservation of momentum of an eulerian fluid: $$\rho\Big(\partial_t\vec{v} + (\vec{v}\cdot\vec{\nabla})\vec{v}\Big) = \rho\vec{b} + \vec{\nabla}p$$
My question is: how that notation could "help" us to understand "how things work"?
I think this is strange, but physicists are known to have strange notations that in some ways help to understand the nature of things. Take for example the divergence of a vector field $\vec{v}\ $: this is written in some texts as $\vec{\nabla}\cdot\vec{v}$ because if we think at $\vec{\nabla}$ as the vector $(\partial_1, \partial_2, \partial_3)$ then the scalar product $$\vec{\nabla}\cdot\vec{v} = (\partial_1, \partial_2, \partial_3) \cdot (v_1,v_2,v_3) = \sum_{i=1,2,3} \partial_i \ v_i = \text{div}(\vec{v})$$
Now, if I take for example a $2$ dimensional field $\vec{v} = (v_1,v_2)$ we have (correct me if I'm wrong) $$L_v(\vec{v}) = \begin{pmatrix} \partial_1 v_1 & \partial_2 v_1 \\ \partial_1 v_2 & \partial_2 v_2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} (\partial_1 v_1)v_1 + (\partial_2 v_1)v_2 \\ (\partial_1 v_2)v_1 + (\partial_2 v_2)v_2 \end{pmatrix} $$
How could I derive "intuitively" the last vector using the notation $(\vec{v}\cdot\vec{\nabla})\vec{v}$?
You already understand that in 2D we are to treat $\overrightarrow{\nabla}$ as $\left\langle \partial_{1},\partial_{2}\right\rangle$. The key for this and many other related notations is basically to treat something like $\partial_1$ as a scalar that's distributive and commutes with scalars (it's a linear operator, after all), and associative where it makes sense, but not commutative.
We have $\overrightarrow{v}\cdot\overrightarrow{\nabla}=\left\langle v_{1},v_{2}\right\rangle \cdot\left\langle \partial_{1},\partial_{2}\right\rangle =v_{1}\partial_{1}+v_{2}\partial_{2}$.
And then: \begin{align}\left(\overrightarrow{v}\cdot\overrightarrow{\nabla}\right)\overrightarrow{v}&=\left(v_{1}\partial_{1}+v_{2}\partial_{2}\right)\left\langle v_{1},v_{2}\right\rangle \\&=\left(v_{1}\partial_{1}\right)\left\langle v_{1},v_{2}\right\rangle +v_{2}\partial_{2}\left\langle v_{1},v_{2}\right\rangle \\&=\left\langle \left(v_{1}\partial_{1}\right)v_{1},\left(v_{1}\partial_{1}\right)v_{2}\right\rangle +\left\langle \left(v_{2}\partial_{2}\right)v_{1},\left(v_{2}\partial_{2}\right)v_{2}\right\rangle \\&=\left\langle v_{1}\left(\partial_{1}v_{1}\right),v_{1}\left(\partial_{1}v_{2}\right)\right\rangle +\left\langle v_{2}\left(\partial_{2}v_{1}\right),v_{2}\left(\partial_{2}v_{2}\right)\right\rangle \\&=\left\langle v_{1}\left(\partial_{1}v_{1}\right)+v_{2}\left(\partial_{2}v_{1}\right),v_{1}\left(\partial_{1}v_{2}\right)+v_{2}\left(\partial_{2}v_{2}\right)\right\rangle \\&=\left\langle \left(\partial_{1}v_{1}\right)v_{1}+\left(\partial_{2}v_{1}\right)v_{2},\left(\partial_{1}v_{2}\right)v_{1}+\left(\partial_{2}v_{2}\right)v_{2}\right\rangle \\&=\begin{bmatrix}\left(\partial_{1}v_{1}\right)v_{1}+\left(\partial_{2}v_{1}\right)v_{2}\\ \left(\partial_{1}v_{2}\right)v_{1}+\left(\partial_{2}v_{2}\right)v_{2} \end{bmatrix}\\&=\begin{bmatrix}\partial_{1}v_{1} & \partial_{2}v_{1}\\ \partial_{1}v_{2} & \partial_{2}v_{2} \end{bmatrix}\begin{bmatrix}v_{1}\\ v_{2} \end{bmatrix}\end{align}