The lapse rate is related to the mathematical slope of the line representing temperature vs. altitude (we have called this the vertical temperature profile in the past). Our definition has the decrease of temperature for increasing altitude as a positive lapse rate; the mathematical slope for a line leaning to the left is negative so we need to put a minus sign in front of the formula for getting lapse rate (also, mathematically, the slope should be dz/dT, but we flip it so it makes more physical sense).

Note that a temperature inversion, where the temperature increases with increasing altitude, has a negative lapse rate. If there was zero lapse rate (the temperature is constant as altitude increases), the temperature profile would be vertical.