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.

Ok, I'm going to assume you recognise the formula for a covariance, but that saying "Cauchy-Schwartz inequality" will not be helpful. So
$$\mathrm{cov}[X,Y]=\frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)(y_i-\bar y).$$
The most similar that $X$ and $Y$ could be would be that they were identical: $y_i=x_i$. In that case, the covariance is
$$\begin{align}
\mathrm{cov}[X,Y]
&=\frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)(x_i-\bar x) =\mathrm{var}[X] \\[6pt]
&=\frac{1}{n-1} \sum_{i=1}^n (y_i-\bar y)(y_i-\bar y) =\mathrm{var}[Y]. \\[6pt]
\end{align}$$
The correlation is then
$$\rho=\frac{\mathrm{cov}[X,Y]}{\sqrt{\mathrm{var}[X]\mathrm{var}[Y]}}=\frac{\mathrm{var}[X]}{\sqrt{\mathrm{var}[X]\mathrm{var}[X]}}=1.$$
Ok, does scaling one of the variables matter? No: if you double $Y$ you end up with a 2 in the numerator and $2^2=4$ inside the square root for $\mathrm{var}[Y]$; the correlation is still 1.
So there's no way you can get the correlation to be bigger than 1, and it's equal to 1 when the two variables are identical or when one is a positive multiple of the other, or (more generally) when one is a positive multiple of the other plus a constant difference -- ie, a straight line relationship.
The most opposite that $X$ and $Y$ could be would be that $Y=-X$: when $X$ goes up, $Y$ goes down by the same amount. The same sort of calculations show that the correlation is then $-1$; it's also $-1$ when one is a negative multiple of the other plus a constant difference -- ie, a straight line relationship
Anything you do to make $(x_i-\bar x)(y_i-\bar y)$ bigger will also make at least one of $(x_i-\bar x)^2$ and $(y_i-\bar y)^2$ bigger. You can make the numerator of the correlation as large as you like, but the denominator will also be large, and the maximum will still be $1$.
