x+y≤-4, so y≤-x-4, so we start by graphing y=-x-4. This is a straight line so we can use the intercepts to draw the line: when x=0, y=-4 (y-intercept) so mark this point on the y-axis); when y=0, x=-4 (x-intercept) so mark this point on the x-axis. Now draw the line through these two points to make the graph of the line.
y is the height of the function -x-4 above the x-axis, so y<-x-4 has to be lower than the line y=-x-4. That corresponds to the whole area to the left of the line, including what lies below the x-axis, so shade that area. y≤-x-4 is either on the line y=-x-4 itself or in the shaded area to the left of the line.