The short version is as follows:
Not realizing this fact has caused numerous fruitless debates.
Example: A pure materialist cannot accept non-material 'stuff' just because it falls outside his view of the world. Any research into that non-material 'stuff' is not considered science. Actually, this is the main reason that most of the 'science vs. believe' debate is not making too much sense. Whether you believe in the existence God or believe that God does not exist is not a scientific issue in a material world, unless one accepts the wider view of the world that has both possibilities in. Pure materialists cannot discuss this question in a sensible way (neither can pure 'believers').
Interestingly, what you don't accept as being part of the real world you can't observe. This is arguably the main reason why progress in science has been so slow in the last 100 or so years.
Example: Newton didn't observe that his prism experiments did not support his theory of colour because he had taken the standpoint, before doing the actual experiments, that colour should behave like tones in music: white light consists of multiple colours that differ in speed with which they travel through a two media, air and the material of the prism. He 'proved' with his experiments that white light did break up in different colours, from red via yellow, green and blue to violet. However, as Goethe showed, this is only true at a certain distance from the prism, and not true at all other distances! Close to the prism one might be able to save Newton, but far away it turns out to be nonsense: we find the sequence red, green, black, violet (see figure 1). For some reason Newton ignored this. Goethe showed that Newton was wrong, and came with an explanation in terms of 'edge colours', i.e. colours are related to each black/white edge that is 'shifted' by the air/prism interface, leading to the different colours. This theory is in exact correspondence with the observable facts
The question then becomes why Newton's theory was adopted, and Goethe's theory ignored. (This is not entirely true, painters cannot work with Newton but love Goethe's...) The reason is twofold: Newton had a very good reputation as a physicist/mathematician, while Goethe was a mere poet, and Newton had cast his theory in mathematical form, while Goethe didn't have the capacity to do so. So, the 'rest of the world' couldn't see that Goethe was right because of their view of the world, not because of the scientific evidence.
(The present-day explanation of the prism experiments is that the human eye is more sensitive to blue than to yellow, which is why we cannot observe the latter some distance from the prism. Actually doing the experiment shows that this explanation is doubtful to say the least. To me, it is totally unclear why scientists cannot accept Goethe's view. Actually, the vast majority of physicists doesn't even realize that there is something fishy with Newton's experiments...)
This question has been the basis of numerous debates in history, and many people much smarter than me have been struggling with it. But, somehow, the importance of the thinking process itself seems to have been overlooked. Let us have a look at how we learn about the world, and concentrate on the thinking process that is involved in this.
a. The first step is the observation. It is good to realize that e.g. a visual observation gives one a two-dimensional picture of the world. However, without thinking about these images, that 'world' is nothing but an incoherent conglomerate of colours and shapes.
b. Only when we become active and think about these colours and shapes we somehow seem to be able to make sense of them. Now that is an extra-ordinary achievement! Were does the idea that certain colours and shapes are connected come from? Maybe it comes from multiple similar observations, and our capacity to order them. That capacity is already part of the thinking process. For instance, one needs to memorize what has been seen before, and be able to remember it at the correct moment. So already in our first encounter with the world we become active and order the observations. Without this ordering process we are unable to make progress.
c. Maybe an aside, maybe not, to be able to order we have to decide what the entities to be ordered are. How do we do that?
d. Apart from the ordering process we also make connections between ordered entities, i.e. this comes before that in space and/or time (let alone where 'space' and 'time' come from).
e. Only then starts the process that we normally denote thinking. Obviously, we have been a bit sloppy here.
f. But there is more. We can also think without any observation interfering. Mathematics is the prime example. We can construct a triangle in our mind and manipulate it, e.g. increase its size and deform it, while keeping it a triangle. We can prove, i.e. convince ourselves (and others) that accepting that a stretched angle is 180 degrees, the sum of the interior angles of a triangle is 180 degrees. And we soon realize that this is true for any triangle. The observation that we can do this, realizing that something is a universal truth, is a very important one. It shows that we can deduce something without interference of objects from the physical world. One might say that the original idea of the triangle came from an observation, but that does not change the fact that the observation did not have this universal truth in it. We, our thinking processes, have brought it in.
g. One than might ask, is this truth part of the real world? A pure materialist has to say no, but I must confess I have a different opinion. One of the arguments in favour is the fact that we can use - and have used extensively - this insight to interfere in the physical world.
h. Actually, this touches upon another quite interesting point related to 'objective science'. It is a phrase used a lot, but asking for its exact meaning gives interesting answers. The most common one is that more than one person has to agree on the facts. Or rather the majority. But that makes science a democratic process, which is not very satisfactorily. Science should tell us about the real world, not about the perception of the majority. (One might accept this democratic view of science, but I think it leads to nothing.) Let us try to shed some light upon this by considering the following experiment.
Assume that a group of people observes a stone being thrown away and upward. It will follow an orbit that resembles a parabola. Now, obviously, each observer in the group has a different standpoint relative to the motion of the stone, so each will observe something different. What is objective about the perception of the motion of the stone? None of the observations has the full truth. Surely, each person will be aware of the fact that a different standpoint gives rise to a different observation. The perception of the true motion is than constructed by combining the different observations. To do this properly needs a lot of thinking! We come to the conclusion that observations themselves are not objective.
On the other hand, those persons in the group that recognized the parabolic shape of the motion of the stone had a parabola in their minds. If they would communicate this among each other they would realize, as we do, that that parabola is quite objective. Furthermore, they might realize, as we do, that the parabola 'belongs' to the actual motion of the stone. Using the idea of the parabola helps interpreting the observation enormously and might be used to construct the 'real' motion of the stone. So our thinking has brought something extra to the pure observation and helps us in reconstructing and understanding what goes on. It is not far fetched, in my view, to say that the thinking process has brought something to the observation that is needed to 'see' the real world. It brings extra knowledge that is not directly given by the observations. It is an essential ingredient in our human way to explore the world. The true world is 'out there', and by our observation process we extract only part of the truth. Our thoughts provide the rest to obtain full knowledge of the real world. Hence, we might conclude with:
Suppose we see a chair in front of us and touch it with our hands. Kant argues that we only have an impression of the chair, but we have no idea what the actual chair is (even, if it is there at all). Present-day physiology seems to support this view. The fotons from the chair stimulate the eye nerves that send an electronic signal to our brain. Where is the chair? The brain receives the signals, analyses them and brings them to our conscience. At the same time nerves in the hand are stimulated by touching the chair, leading to a similar process. But how does the touching work? In a molecular view of the world touching is a complicated business. And considering the nerves, it is well known that one can stimulate visual images by stimulating the eye nerve mechanically, i.e. the signals in the nerves do only have limited information on the actual 'image' they transfer. Given all these transformations of the characteristics of the chair, how on Earth can we think that we have the real chair in our conscience?
This 'solid reasoning' has at least one weak point. What do we know about our eye, our nerves, our hand, our brain? If we try to be consistent with Kant, we only have impressions of these. But what does the reasoning above then mean? Indeed, not much. Kant has not proved that we can only have impressions of the real world, but never find the real world itself. His argument is false, and will not influence our reasoning any further.