There are already several good answers here, including in the comments. However as the OP requested a "simpler" justification, here I will expand on my comment.
To me this is a very natural distinction between root-mean-square vs. mean-absolute deviations, and why we might prefer one vs. the other when measuring dispersion. (I do not know if it is "simpler"?)
Say you have some data $x_1,\ldots,x_n$, which you want to approximate by a constant $c$, i.e. $$x_i\approx c$$ for all $i$.
How do you choose the constant? A common approach is to minimize some error $E[c]$.
One choice for $E$ is the sum square error $$E_\text{SSE}=\sum_i\big(x_i-c\big)^2$$ the solution will then be $c_\min=\frac{1}{n}\sum x_i$. In other words, we have $$\big[c_\min,E_\min\big]_\text{SSE}=\big[\text{mean}(\mathbf{x}),n\,\text{var}(\mathbf{x})\big]$$ so if you are using the mean as your measure of central tendency, the RMS error is really the "natural" measure of dispersion.
On the other hand, if we choose $E$ to be the sum absolute error $$E_\text{SAE}=\sum_i\big|x_i-c\big|$$ the solution will then be $(c_\min)_\text{SAE}=\text{median}(\mathbf{x})$. So if you want to use mean absolute deviation to measure dispersion, really the "natural" measure of central tendency would be the median.
Summary: If you want to use mean absolute deviation, then arguably you should be measuring dispersion around the median. If you are already using the mean, then arguably standard deviation is the appropriate measure of dispersion. Here "arguably" is justified by optimality (minimum dispersion).