I read the article Wonderful Models of Subspace Arrangements (https://link.springer.com/content/pdf/10.1007/BF01589496.pdf) and have the trouble with the following fragment:
Here $K$ is a field. What here we mean by coordinates on vector space $K^E$?
How map $\rho_{\mu}$ acts on vector?
Why we need the condition ``the elements greater than any given one a form a linearly ordered set'' to define the inverse map?
Let's do a simple example, and then everything should become clear. Let $K=\Bbb R$ and $E=\{1,2,3\}$ with the obvious order. Write $x_1,x_2,x_3$ for coordinates on the source copy of $\Bbb R^3$ and $y_1,y_2,y_3$ for coordinates on the target copy. Then we have that $y_i=\prod_{j\geq i} x_j$, so our map $\rho$ sends a vector $(x_1,x_2,x_3)$ to the vector $(x_1x_2x_3,x_2x_3,x_3)$. The inverse is then $(y_1,y_2,y_3)\mapsto (\frac{y_1}{y_2},\frac{y_2}{y_3},y_3)$ which indeed gives us an inverse on the open dense set where $y_2,y_3\neq 0$.
Now let's handle your questions in the general case. For the coordinates, since $K^E$ is the set of maps $E\to K$, one very natural way to pick coordinates on this space would be for an element $v$ of the vector space $K^E$ (a map $E\to K$) to associate to every $a\in E$ the image of $a$ under the map $v:E\to K$. Thinking it through, this just means that elements of $K^E$ are vectors where the entries are indexed by the elements of $E$, which matches up with our usual interpretation of $K^n$ for some integer $n$.
The definition of $\rho$ tells you that given an input vector $v$ with coordinates $u_b$ as $b$ ranges through $E$, the map $\rho$ sends this vector to the vector $w$ which has as it's $a$-coordinate $\prod_{b\geq a} u_b$. See, for instance, the example above.
The construction of the inverse requires that every element have either zero or one elements which are minimally properly greater than it. Since $E$ is finite, this is equivalent to $E$ being linearly ordered: every totally ordered subset of size $n$ can be put in an order-preserving bijection with $[1,2,\cdots,n]$, which should be equivalent to the definition of a linearly ordered finite set (if you have a specific definition in mind, please add it to your post).