The short answer is: BECAUSE THE ICE IS BLUE.
Now we have to explain why it seems perfectly transparent on ice cubes and industrial ice blocks.
It has to do with the fact that most transparent materials are not perfectly transparent, and instead absorb (and/or scatter) part of the light that hits them. And when the transparency is better for one specific color (wavelength), the material will look that color, much like how stained glass gives a tint to the white light going through it.
Now the important fact is that this "tint" effect is stronger the thicker the material is. Let's take a standard glass as a example: When looking through the glass it seems perfectly transparent, because the thickness is too small to make its color noticeable. However, if you look at the edge of the glass, it looks green, which shows that glass is indeed green, but that only became noticeable when looking through a section thick enough.
In the same way, glacier ice is blue, but that can only be appreciated if the piece of ice you are looking at is thick enough. We can see this effect in the following picture of an iceberg, where sections of different thicknesses displays a color gradient ranging from transparent to blue.
The remaining questions is: Why are glaciers then not always blue? As we know many icebergs (and deep snows) are white.
The key here is bubbles. The reason is that for each ice/air interface, part of the light gets reflected and only a fraction goes through. Therefore the greater the number of bubbles a piece of ice has, the less light will be able to proceed deeper into the ice. As such, when you view light from more bubbly ice, most of it will have traveled through less ice, and thus have had less distance to obtain the blue tint. I've made this figure to help understanding the idea:
On the left, a piece of ice with very few bubbles allows a longer travel path through the ice, inducing a marked blue tint in the outgoing light. On the right, in contrast, if there are many bubbles the light will go bounce back right away with only a very subtle tint, or no tint at all.
Snow is the extreme of this, because it is mostly air (bubbles), and only some small pieces of ice, that's why it looks so white.
An interesting ramification of this: those deep blue pieces of ice started as snow. But as the snow piles up, the weight of the snow on top compacts it, squeezing air out changing it from snow to firn until you finally reach the density of being a block of ice (with bubbles). In a big glacier, hundreds or thousands of meters of ice can exert such a pressure that the bubble spaces get compacted smaller and smaller until they disappear and the air is dissolved into the ice itself.
Therefore this means that the more blue (and bubble-less) a piece of ice is, the more it has come from deeper within the glacier... meaning it also has traveled from further up in the glacier basin and it is, therefore, older. So, now you can identify deep, old ice just by its color.
To give a more familiar example of how adding bubbles can turn something transparent with a tint into something white, take the case of egg whites, which are transparent with a yellowish tint. But as you whip it, adding in bubbles, it begins to turn into meringue which is perfectly white.
However, there is a trick that can even make the blue color of seemingly-white snow noticeable. If you let a ray of light bounce on it many times, each time it will travel through a little bit of ice, getting a bit of blue tint each time, and after enough bounces it will look unquestionably blue. That's what you see when looking at the entrance of a snow cave or a crevasse in a glacier. The light that comes from deeper in the cave has bounced many more times around the walls and becomes bluer: