Preface
I struggled with what Landau writes in the section on Mechanical similartiy [1]. Like the OP, "I [too] was looking for some more detailed explanations of the matter, at a level like the one in the first chapters of Landau's book." I looked at a few places including [2]. I was not satisfied with what I found. So, I prepared my own detailed explanation, which I set down below. In this answer, I quote liberally and without attribution from Landau and Lifshitz; and some Wikipedia pages.
Answer
My answer is divided into three major components. In the first component, I present a short summary of an essential proposition that the reader should be familiar with by the time they reach the section on Mechanical similarity in Landau [1]. In the second component, I review what is meant of when we speak of homogeneous functions; and I present two examples. The results of both these examples are used in the third component. In the third component I first present Proposition 2 that regards what is means when we speak of mechanical similarity; and next, I prove that proposition. Equation (5) in Proposition 2 describes the exact similitude. Bear in mind that Equation (5) is not identical to Equation (10.2) given by Landau and Lifshitz in their explanation of mechanical similarity [1].
First Component
Proposition 1: Multiplication of the Lagrangian by any constant does not effect the equations of motion.
Proof.
\begin{align*}
\left(\frac{d}{dt} \frac{\partial [kL]}{\partial{\dot{q}} } \right)
-\frac{\partial [kL]}{\partial{ {q}} } =0
\\
k\left(\left(\frac{d}{dt} \frac{\partial [L]}{\partial{\dot{q}} } \right)
-\frac{\partial [L]}{\partial{ {q}} }\right) =0
\end{align*}
Q.E.D
This proof makes possible, in a number of important cases (cf, scattering Problem 3 [2]) some useful inferences concerning properties of the motion, without the necessity of actually integrating the equations of motions. These inferences rely on the properties of homogeneous functions.
Second Component
Definition [Cone, linear cone]
A subset $C$ of a vector space $V$ over an ordered field $F$ is a cone (or sometimes called a linear cone) if for each $x$ in $C$ and positive scalar $\alpha$ in $F$, the product $\alpha x$ is in $C$.
Example 1:
The set $$\left \{ x \in \mathbb{R}^2 \mid x_2 \geq 0, x_1 = 0 \right \} \cup \left \{ x \in \mathbb{R}^2 \mid x_1 \geq 0, x_2 = 0 \right \}$$ is a cone, but not a convex cone.
Example 2:
The norm cone $$ \left \{ (x, r) \in \mathbb{R}^{d+1} \mid \|x\| \leq r \right \}$$
is a convex cone.
Definition [Homogeneous function]
Let $V$ and $W$ be two vector spaces over a field $F$. A homogeneous function $f$ from $V$ to $W$ is a function from $V$ to $W$ that has a linear cone $C$ as its domain, and satisfies
$$ f(s\,x)=s^{k}f(x)$$
for some integer $k$, every $x \in C$, and every nonzero $ s\in F$. The integer $k$ is called the degree of homogeneity, or simply the degree of $f$.
Example 3:
The kinetic-energy function, $T : \mathbb{R}^{3n} \to \mathbb{R}$ is the rule that assigns to each tuple
$\left(\dot{\mathbf{q}}_1,\ldots, \dot{\mathbf{q}}_n\right)$ in $\mathbb{R}^{3n}$ the value
$$T{\left(\dot{\mathbf{q}}_1,\ldots, \dot{\mathbf{q}}_n\right)} = \sum\limits_{i=1}^n k_i \,\dot{\mathbf{q}}_i^2 $$ in $\mathbb{R}$. The kinetic-energy function $T$ is homogeneous of degree $2$:
$$T{\left(\frac{\alpha}{\beta}\,\dot{\mathbf{q}}_1,\ldots,\frac{\alpha}{\beta}\,\dot{\mathbf{q}}_n\right)} = \sum\limits_{i=1}^n k_i \,\left(\frac{\alpha}{\beta}\,\dot{\mathbf{q}}_i\right)^2 = \left(\frac{\alpha}{\beta}\right)^2\sum\limits_{i=1}^n k_i \,\left( \dot{\mathbf{q}}_i\right)^2 = \left(\frac{\alpha}{\beta}\right)^2 f{\left( \dot{\mathbf{q}}_1,\ldots, \dot{\mathbf{q}}_n\right)}.$$
Example 4:
Let $k$ be an element in the field $\mathbb{R}$. Let $j$ be an element in the field $\mathbb{N}$.
The family of central field potential-energy functions, $V_k : \mathbb{R}^{3n} \to \mathbb{R}$ is the rule that assigns to each tuple
$\left( {\mathbf{q}}_1,\ldots, {\mathbf{q}}_n\right)$ in $\mathbb{R}^{3n}$ the value
$$
V_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n\right)} = \sum_{i=1}^{n}\sum_{l=i+1}^{n}\left|\mathbf{q}_i- \mathbf{q}_l\right|^k
$$
in $\mathbb{R}$.
The family of central field potential-energy functions $V_k $ is homogeneous of degree $k$:
$$
V_k {\left(\alpha\mathbf{q}_1,\ldots, \alpha\mathbf{q}_n\right)}
=
\sum_{i=1}^{n}\sum_{l=i+1}^{n}\left|\alpha\mathbf{q}_i- \alpha\mathbf{q}_l\right|^k
=
\alpha^k\sum_{i=1}^{n}\sum_{l=i+1}^{n} \left| \mathbf{q}_i- \mathbf{q}_l\right|^k
=
\alpha^k\,V_k {\left( \mathbf{q}_1,\ldots, \mathbf{q}_n\right)}
\,.$$
Third Component
Proposition 2 [Exact mechanical similitude]
Let $k$ be a fixed element in the field $\mathbb{R}$. Let $n$ be a fixed element in the field $\mathbb{N}$.
Consider the Lagrangian of a conservative system,
$$L_k
= L_k{\left( \mathbf{q}_1,\ldots, \mathbf{q}_n ,
\dot{\mathbf{q}}_1,\ldots, \dot{\mathbf{q}}_n \right)} \,,
$$
where the potential energy is a central field given by the homogeneous power law
$$
V_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n\right)} = \sum_{i=1}^{n}\sum_{j=i+1}^{n}c_{i,j}\left|\mathbf{q}_i- \mathbf{q}_j\right|^k \,.
$$
The trajectories (i.e., $\mathbf{q}_1(t),\ldots,\mathbf{q}_n(t)$) that satisfy the equations of motion generated by the Lagrangian have the property of exact similitude. Namely, for any $t \in \mathbb{R}$, any $i= 1,\ldots,n$, any $j = 1,\ldots, n$, every $\mathbf{q}_i \in \mathbb{R}^3$, every $ \mathbf{q}_j \in \mathbb{R}^3$, any integer $k$, and every nonzero $ \alpha \in \mathbb{R}$, the exact mechanical similitude is given by the equation
$$ \left| \alpha\,\mathbf{q}_i {\left( \alpha^{\left[1-\frac{k}{2}\right]}\,t\right)} -
\alpha\,\mathbf{q}_j { \left( \alpha^{\left[1-\frac{k}{2}\right]}\,t\right) } \right| = \alpha \left| \mathbf{q}_i{\left( t\right)} - \mathbf{q}_j{\left(t\right)} \right|\,\tag{5}.
$$
Proof. Let $\alpha$ and $\beta$ be scalars such that $\alpha \in \mathbb{R}$ and $\beta \in \mathbb{R}$. Let $f: \mathbb{R}^3 \to \mathbb{R}^3$ be the rule assigns to each $\mathbf{q} \in \mathbb{R}^3$ the value
$$\mathbf{q}' = f(\mathbf{q}) = \alpha\,\mathbf{q}\, \tag{10}$$ where
$\mathbf{q}'\in \mathbb{R}^3$.
Let $h: \mathbb{R} \to \mathbb{R}$ be the rule assigns to each $t \in \mathbb{R}$ the value
$$t' = h(t) = \beta\,t\,,\tag{15}$$
where
$t'\in \mathbb{R}$.
Let $h^{-1}: \mathbb{R} \to \mathbb{R}$ be the rule assigns to each $t' \in \mathbb{R}$ the value
$$t = h^{-1}(t') = \frac{1}{\beta}\,t'\,,\tag{20}$$
where
$t\in \mathbb{R}$.
Since $\mathbf{q}' = f(\mathbf{q})$, $\mathbf{q} = g(t)$, and $t = h^{-1}(t')$, then the chain rule is written in Leibniz notation as:
\begin{align*}
\frac {d\mathbf{q}'}{dt'}
&=
\frac {d\mathbf{q}'}{d\mathbf{q}}\,\,\frac {d\mathbf{q} }{dt} \,\,\frac {dt}{dt'} && \text{chain rule}
\\
&=
\alpha\,\frac {d\mathbf{q} }{dt}\,\frac {dt}{dt'} && (10)
\\
&=
\alpha\,\frac {d\mathbf{q} }{dt}\,\frac {1}{\beta} && (20)\\
&=
\frac {\alpha}{\beta}\,\frac {d\mathbf{q} }{dt} && \text{commutation}\,.
\end{align*}
Thus,
\begin{align*}
&L_k{\left(\mathbf{q}'_1,\ldots, \mathbf{q}'_n, \frac{d\mathbf{q}'_1}{dt'},\ldots, \frac{d\mathbf{q}'_n}{dt'} ,t'\right)}
\\
&\quad=
L_k{\left(\alpha\,\mathbf{q}_1,\ldots, \alpha\,\mathbf{q}_n, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_n}{dt} ,\beta\,t \right)}
\end{align*}
Yet,
\begin{align*}
&L_k{\left(\alpha\,\mathbf{q}_1,\ldots, \alpha\,\mathbf{q}_n, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_n}{dt} ,\beta\,t \right)}
\\
&\quad =
T{\left( \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_n}{dt} \right)}
-
V_k{\left(\alpha\,\mathbf{q}_1,\ldots, \alpha\,\mathbf{q}_n \right)}
&&\text{conservative system}
\\
&\quad =
\left(\frac{\alpha}{\beta}\right)^2T{\left( \frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{d\mathbf{q}_n}{dt} \right)}
-
V_k{\left(\alpha\,\mathbf{q}_1,\ldots, \alpha\,\mathbf{q}_n \right)}
&&\text{Example 3}
\\
&\quad =
\left(\frac{\alpha}{\beta}\right)^2T{\left( \frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{d\mathbf{q}_n}{dt} \right)}
-
\alpha^k \, V_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n \right)}
&&\text{Example 4}
\end{align*}
Landau's premise is that $\beta = \alpha^{1-\frac{k}{2}}$. Therefore,
\begin{align*}
&L_k{\left(\alpha\,\mathbf{q}_1,\ldots, \alpha\,\mathbf{q}_n, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{\alpha}{\beta}\,\frac{d\mathbf{q}_n}{dt} ,\beta\,t \right)}
\\
&\quad =
\left(\frac{\alpha}{\alpha^{1-\frac{k}{2}}}\right)^2T{\left( \frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{d\mathbf{q}_n}{dt} \right)}
-
\alpha^k \, V_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n \right)}
\\
&\quad =
\alpha^k\left( T{\left( \frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{d\mathbf{q}_n}{dt} \right)}
-
V_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n \right)}\right)
\\
&\quad =
\alpha^k\,L_k{\left(\mathbf{q}_1,\ldots, \mathbf{q}_n , \frac{d\mathbf{q}_1}{dt} ,\ldots, \frac{d\mathbf{q}_n}{dt},t \right)}\,.
\end{align*}
From Proposition 1, since the transformations given in Eqs. (10) and (15) (in conjunction with the premise that necessitates that $\beta = \alpha^{1-\frac{k}{2}}$) produce a constant multiple of the Lagrangian, therefore these two transformations do not effect the equations of motion.
From this we can understand that the trajectories that satisfy the equations of motion generated by the Lagrangian will have the property of exact similitude [3]. Namely, the exact similitude is given by the rule in the equation
$$ \left| \overbrace{\mathbf{q}'_i}^{\alpha\,\mathbf{q}_i}\underbrace{\left( t'\right)}_{t\,\alpha^{\left[1-\frac{k}{2}\right]}} -
\overbrace{ \mathbf{q}'_j}^{\alpha\,\mathbf{q}_j}{\underbrace{\left(t'\right)}_{t\,\alpha^{\left[1-\frac{k}{2}\right]} }} \right| = \alpha \left| \mathbf{q}_i{\left( t\right)} - \mathbf{q}_j{\left(t\right)} \right|\,.
$$
This rule is valid for any $i,j=1,\ldots,n$ and for any time $t$.
Q.E.D.
Bibliography
[1] Landau, Volume 1, 3rd Edition, pp 22--23, 51.
[2] Wikipedia contributors. Mechanical similarity [Internet]. Wikipedia, The Free Encyclopedia; 2021 May 24, 09:26 UTC [cited 2022 Dec 12]. Available from: https://en.wikipedia.org/w/index.php?title=Mechanical_similarity&oldid=1024830575.
[3] https://en.wikipedia.org/wiki/Similarity_(geometry)
