In my reading, I came across the following ODE system:
$$\lambda_1 \dot x = f(x,y)$$ $$\lambda_2 \dot y = g(x,y)$$, where $\lambda_1$ and $\lambda_2$ are positive constants.
Then, I saw that the authors call $\lambda_1$ and $\lambda_2$ "relaxation constants". But, what does that mean? I mean, what does these constants do and what do they "relax"?
Relaxation is an adjustment of the speed of an iterative process that makes the updates more gentle (under-relaxation) or more aggressive (over-relaxation). A prototypical example of over-relaxation is described on Wikipedia. An example of under-relaxation is in this answer.
In your context, it's only the ratio of constants $\lambda_1/\lambda_2$ that really matters, since multiplying them both by the same factor does nothing but change the time scale. One could just as well use only $\lambda_2$ (this is what is done in the gentlest ascent article).
Making $\lambda_2$ large compared to $\lambda_1$ will make $y$ change its value more slowly compared to $x$, and vice versa. The effect on the speed and stability of convergence depends on what $f$ and $g$ are. Typically one carries out the analysis for general $\lambda_{1,2}$ and then picks the values that help achieve desired estimates at the end.
As an illustration, one can think of mass in Newtonian dynamics as a relaxation coefficient: $$ \begin{split}\dot x &= v \\ m\dot v &= F(x) \end{split} $$ Here $x$ and $v$ are the position and velocity of a particle of mass $m$. The coefficient $m$ determines how quickly the particle will change its velocity under the action of the force. Depending on the geometry of the force field, this can have a complex effect on dynamics.
Edgar Allan Poe described something of the sort in his story A Descent into the Maelström.