As with all word problems, once you translate the sentences into equations it becomes quite straightforward. Let's go through each, letting h be the height, l the length, w the width, and V the volume.
The height of a box is twice it's length: h = 2l
The length of the box is twice its width: l = 2w
note that you may substitute the second equation into the first to get: h = 2(2w) = 4w
And finally the harder sentence. (note: I assume you meant the volume is increased by 71 cubic inches). We will use V as the box's original volume. You should already know that the volume of a box is described by the equation V = lwh. To transform this sentence to an equation, simply add 1 to each dimension and 71 to the total volume:
(l+1)(w+1)(h+1) = V+71 = lwh+71
Now with three equations and three unknowns, you should be able to carry the problem through on your own. However, as you requested a full solution, I will continue. Let's solve with substitution, using the first two equations to get everything in terms of w.
(2w+1)(w+1)(4w+1) = (2w)(w)(4w)+71 = 8w3+71
multiply out the left side to get:
8w3+14w2+7w+1 = 8w3+71
subtract from both sides to get:
14w2+7w-70 = 0
7(2w2+w-10) = 7(w-2)(2w+5) = 0
note that for that equation to be true, either w-2 = 0 or 2w+5 = 0. Solving those two equations gives us w = 2 or w = -5/2. We can safely throw out the negative answer, as in our original context it won't make sense. Now we simply use w = 2 in our first two equations to get our full answer:
w = 2
l = 2w = 2(2) = 4
h = 2l = 2(4) = 8
And as always lets check to make sure this is correct by calculating the volume:
V = lwh = (4)(2)(8) = 64
(l+1)(w+1)(h+1) = (5)(3)(9) = 135 = 64+71
So everything checks out!