Suppose you work for NASA and are the person who is given the task of announcing the time to the launch of a particular rocket. You will call this time $t=0$ and so just five seconds before the launch you will announce "t minus 5..4..3..2..1..ignition". Now this time $t=0$ could have been for example 28th February 12 noon, and everything before that particular event exists on the "negative time" axis, or all points in time before this event.
We do the same thing in classical physics. We can define any point to be $t=0$ and it simply defines what will happen/happened at this point, and everything prior to this point is the past and everything after it is the future.