With my very limited knowledge of development economics:
$\left(\frac{x_m}{x}\right)^\alpha$ represents the proportion of the population that has an income larger or equal to $x$ where $x\geq x_m>0$ and $x_m$ is the minimum income amount.
Example 1:
Suppose $\alpha\rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $50,000$?
$$\lim_{\alpha \rightarrow 1} \left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{50,000}\right)^1=1$$
This suggests that $100\%$ of the population has an income larger than or equal to $50,000$.
Example 2:
Suppose $\alpha \rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?
$$\lim_{\alpha \rightarrow 1}\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^1=0.1$$
This suggest that $10\%$ of the population has an income larger than or equal to $500,000$.
The story does not end there. What about $\alpha$? It must be the case that $\left(\frac{x_m}{x}\right)^\alpha$ is bounded between 0 and 1 (it is a proportion). As such, it must be the case that $\alpha>1$. In the literature, $\alpha$ is refered to as the Pareto Index. Let us consider another example.
Example 3:
Suppose $\alpha=2$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?
$$\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^2=0.01$$
This suggest that $1\%$ of the population has an income larger than or equal to $500,000$. From this, a high $\alpha$, or a high Pareto Index, suggests a small proportion of high-income individuals.
What is a reasonable value of $\alpha$? We can normalize $x_m = 1$. Suppose we wanted to know the share of total income that is received by those individuals with an income above, say $\tilde{x}$. We take the following integrals:
$$\int_{x_m}^{\infty} x^{-\alpha} dx = \int_{1}^{\infty} x^{-\alpha} dx = \frac{1}{\alpha-1}$$
$$\int_{\tilde{x}}^{\infty} x^{-\alpha} dx = \frac{\tilde{x}^{1-\alpha}}{\alpha-1}$$
Therefore the share of total income that is received by those individuals with an income above $\tilde{x}$, call it $q$, is the ratio of the two integrals:
$$q = \tilde{x}^{1-\alpha}$$
We also want to define the proportion of the high-income population with an income larger than $\tilde{x}$ as $p$:
$$p = \tilde{x}^{-\alpha}$$
The next logical question would be what value of $\alpha$ allows us to reconcile the empirical observation that $20\%$ of the population has $80\%$ of the total income? This implies that $p=.2$ and $q=.8$.
$$q = p^{\frac{\alpha-1}{\alpha}}$$
Taking the log of both sides:
$$\ln(q) = \frac{\alpha-1}{\alpha}\ln(p)$$
Solving for $\alpha$:
$$\alpha = \frac{\ln(p)}{\ln(p)-\ln(q)} = \frac{\ln(.2)}{\ln(.2)-\ln(.8)} = 1.16$$
Therefore, an $\alpha=1.16$ is consistent with what is observed empirically. To answer the question, $x_m$ is the minimum income amount and is usually normalized to $1$. $\alpha$ is the Pareto Index which is a measure of the proportion of high-income individuals. A high $\alpha$ suggests a low number of high-income individuals.
The transformation $L(F)$ simply allows you to compute what % of the total wealth is controlled by those with an income below $\tilde{x}$.
