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 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 always in classical mechanics. 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.

Consider also a case where we have a position versus time graph, which shows an object with displacement according to the equation $$x(t)=(5t^2+1) m$$ where $m$ represents metres and $t$ is measured in seconds. You are told its displacement at say $t=1s$  is $6m$. You can confirm this by substituting $t=1s$ into this equation. But now you want to know what was it's displacement at $t=-2 s$. You can once again substitute this value for $t$ into the same equation and you will know that the object had a displacement of $21m$, two seconds before it reached the axis defined by $t=0s$.

"Negative time" simply means all points in time before a specific event, that we say will happen at $t=0s$.