Why is the correlation coefficient always smaller than $1$ and bigger than $-1$? I kind of see that the covariance is similar to the product of the standard deviations but I can't figure out how. My problem is that I don't understand how does this formula tells us how strong the connection is between 2 variables.

The answer by Thomas Lumley gives an intuitive view of the matter, so I'm going to show it using vector notation via the law of cosines to give an alternative view of the matter.  To do this, suppose we create vectors $\mathbf{u}=(u_1,...,u_n)$ and $\mathbf{v}=(v_1,...,v_n)$ where the elements are deviations from the means:
$$u_i = x_i-\bar{x}
\quad \quad \quad \quad 
v_i = y_i-\bar{y}.$$
We can now write the sample covariance and the sample variances in vector notation as:
$$\begin{align}
r_{X,Y} &= \frac{1}{n-1} \sum_{i=1}^n u_i v_i = \frac{1}{n-1} \mathbf{u} \cdot \mathbf{v} \\[6pt]
s_{X}^2 &= \frac{1}{n-1} \sum_{i=1}^n u_i^2 = \frac{1}{n-1} ||\mathbf{u}||^2 \\[6pt]
s_{Y}^2 &= \frac{1}{n-1} \sum_{i=1}^n v_i^2 = \frac{1}{n-1} ||\mathbf{v}||^2 \\[6pt]
\end{align}$$
The law of cosines says that $\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \ ||\mathbf{v}|| \cos \theta$ where $\theta$ is the angle between the vectors.  This allows us to write the (Pearson) sample correlation coefficient as:
$$\rho_{X,Y} 
= \frac{r_{X,Y}}{s_X s_Y} 
= \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| \ ||\mathbf{v}||}
= \cos \theta.$$
That is, the sample correlation is equivalent to the cosine of the angle between the vectors of deviations from the sample means for the two samples.  Now, we know that $-1 \leqslant \cos \theta \leqslant 1$ for all $\theta \in \mathbb{R}$, so this establishes the range restriction.
Incidentally, this interesting mathematical rule is just a basic aspect of the geometric analysis of random variables.  Some broader geometric properties of the linear relationships between multiple variables are examined in O'Neill (2019).  In that paper you will see that both the Pearson sample correlation and the broader coefficient of determination are fully determined by the cosines of the angles between the vectors of deviations of the variables of interest.  This means that regression analysis (encompassing measurement of Pearson correlation for two variables) can be conceived in relatively simple geometric terms, where key measures of goodness-of-fit are scale free measures based on the cosines of the angles between the vectors of deviations of the variables.
