So we can think of the plot of #y = |x|#:
graph{|x| [-10, 10, -2, 12]}
We want this to shift six units left, so it has to look like
graph{|x+6| [-10, 10, -2, 12]}
So we see really easily that the corner that used to be at 0 is now at #-6#. Also, the value at 0 is now 6. We can't just do something like #|x| + c#, since that would move the function up. Instead, if we put a number in the absolute value, it might do what we want.
So let's imagine #|x+c|#. At #x=0#, we get #|c|#, so we know that #c = pm 6#. We can now easily see which is it by plugging in #x=-6#. That should give us 0, and only does that when #c = 6#. Therefore, we get
# y = |x+6|#
which is what I plotted in the second equation.
The way that I think of it is that you have to get to #x = 0# more quickly, so you have to add things. The numbers start "looking ahead" relative their normal values, which is why it shifts left and not right despite adding a number.
