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.

Quick tip to commit to memory: $\text{Correlation}(X, Y)$ = the expected cosine angle between centered versions of $X$ and $Y$.
Here the centering is done by subtracting off the expected value of each random variable from itself.
