The term "-ve edge" or negative edge is shorthand referring to one part of the typical rectangular waveform (or 'square wave') at which the voltage is falling from a high value to a low value. Since this change is typically very fast in a digital circuit, it is referred to as an edge. You might also hear the other term "positive edge". The positive and negative edges are often also referred to as the rising and falling edges or even the leading and trailing edges.
To differentiate its output is a mathematical reference. If you have studied basic calculus, you will have heard of the term "to differentiate". It really means to calculate the instantaneous slope of a value which is assumed to be changing. I hope you know what slope means. You will learn as you go into electronics that mathematics is used to describe the behavior of electronic circuits because those circuits are extremely predictable using such mathematics. So learn as much math as you can manage. Its all useful in electronics. When he says to differentiate the signal, he refers to using a simple circuit which "takes the differential of" the signal. The most common circuit to do this is with a series connection of a capacitor followed by a shunt connection of a resistance. We often refer to such simple configurations as an RC circuit for obvious reasons. To explain further needs a schematic so, being lazy, I will stop here.