Importance of having a "connection" while applying to PhD
At least for schools in the US, it is not necessary for you to have a direct "connection" or contact at a department to be admitted.

In general, where contacts can be useful is if one of the people who writes your letters of recommendations for you knows a particular person in the given department. If that "connection" is good, and you get a good letter of recommendation, that will carry more weight than a typical letter of the same quality, since you will be better "known" than someone who the department doesn't know at all.

That said, though, there is no expectation that you would directly know someone already at the school—although your research interests should align with the department as a whole.