How to understand clearly the feature importance computing in random forest model
Let's start with this formula for the Variable Importance Measure (VIM) based in Gini Impurity (GI)*: 

$$VIM_{jm}^{(Gini)} = GI_m - GI_l - GL_r$$

Here, $m$ is a given node in a single decision tree which performs a split on feature $j$. For this node you can compute the Gini Impurity $GI_m$ (see * note at the very bottom for the formula). Now this node has two childs: a left node $l$ and a right node $r$. For these the Gini Impurities are $GI_l$ and $GL_r$ respectively. If you substract the Gini impurities of the child nodes from the Gini Impurity of node $m$, then you know by how much the split performed by this node has decreased the overall Gini Impurity. Because before the split it was $GI_m$ and after the split the remaining impurity is the sum of $GL_l$ of $GL_R$. Therefore, the difference ($GI_m - (GI_l + GL_r) 
= GI_m - GI_l - GL_r$) is the corresponding impurity reduction. 

(as a side node: this is the unweighted gini importance calculation since $GI_l$ and $GI_r$ are considered with equal weights. There are also different weighted versions. For details and a comparison please refer to "The effect of splitting on random forests", Hemant Ishwaran, Mach Learn (2015) 99:75–118)

For feature importance however, you're interested in the overall importance of each feature and not a single node. And a given feature can be present in different branches of a tree. 

Here is an example:



(Source: "Artificial intelligence - a modern approach"; Norvig, Russell; 2010; 3rd Ed)

As you can see "Alternate?" is being used at multiple nodes. That is why they use set $M$ in the blog you posted to derive the overall importance of a feature $j$ in a tree $i$:

$$VIM_{ij}^{(Gini)} = \sum_{m \in M}VIM_{jm}^{(Gini)}$$

As a side node: in some papers and books the authors do not calculate the impurity indexed by features $j$ but indexed by a split value of a given node, e.g. $s$, (which of course operates on features). Don't get confused by that.   



*
The Gini Impurity of a given node $m$ with $k \in K$ being the classes is defined as follows:
$$GI_m = \sum_{k \in K} \hat{p}_k(m)(1- \hat{p}_k(m)).$$

Where $\hat{p}_k(m)$ is the class frequency of class $k$ in node $m$.