Why does water cast a shadow even though it is considered 'transparent'?

A large amount of water (i.e. if the path of light through it is long) will simply start absorbing light, as it's not completely transparent. For smaller amounts, as when pouring it from one container to another, this is mostly negligible. However, there is also surface reflection. A small amount of the incident light will be reflected off by the surface.

The much larger contribution, however, will come from refractive effects. If you look closely, there are not only areas which are darker than the uniformly lit surroundings, but some will also be brighter. The stream of water forms shapes that act similar to a lens and will divert light off its original path. The patches where the incident light would have gone without the disturbance will then be darker, the places where it is directed to instead will be brighter.

This is just to add an illustration to noah's and Ralf Kleberhoff's answers which correctly point out that refraction is the main reason.

Illustration of refraction-shadow

Note that although most of the light rays do make it through the water drop, most of them do not continue on the path with the rest of the light bundle, but end up somewhere else. As a result, right behind the drop, the light intensity is much lower than it would be without the drop, whereas around this shadow it'll actually be slightly higher.

Although there are already some good answers, I'd like to give it another attempt.

Water is transparent in the sense that most of the light that enters some volume of water, also exits at the other end (unless we talk about multiple meters of thickness or lots of dirt in the water). But that doesn't mean that it passes water unchanged.

Wherever light enters water from air, it changes its direction unless the entry is strictly perpendicular to the water surface at entry point. That's called refraction, the same effect that's used with glass instead of water to form optical lenses. And refraction also happens at the exit point.

When you pour water from a container, the stream of water will typically have quite an irregular shape. So it's more than likely that every single ray of light changes its direction at entry and exit, and only a small portion will continue in a similar direction compared to entering the water stream. So, the light entering the water stream gets distributed over a much larger area than it were without the water, meaning that the average intensity is accordingly lower. And this distribution isn't uniform, but iregular because of the water stream's shape.

Now what is a shadow? It's the darker area behind some object that isn't reached by direct sunlight, while the surroundings are exposed to the sun (of course that works with any similar light source as well).

It's very similar with the water stream's shadow, it is the area that isn't reached by direct sunlight, and the refracted light after passing through the water doesn't compensate for that.

The visible difference is that the refracted light creates irregular patterns of (dim) light behind the water (because of its irregular shape), both in the "shadow" area as well as in its vicinity, because the light hitting the water isn't completely lost, but refracted.

In addition, even if the water had a completely regular planar shape perpendicular to the light source, so that the light rays continue in exactly their original direction, you'd still see a "shadow" area less brightly lit than its surroundings. And that's caused by reflection, as, when crossing a media boudary, some part of the light doesn't enter the other medium, but gets reflected backwards.