MySQL, like most databases, by default runs locally and disallows access from outside networks. As such, you cannot connect to it from an external computer.
Navicat, being a software explicitely for remote administration of databases, likely connects via SSH and tunnels the MySQL connection over it. That way it can act as if the database was installed locally, and for the database it looks as if it was accessed locally.
You could try to do the same by creating a tunnel using Paramiko; see also this question.