Nevertheless, isn't Kant making an assumption by saying there are "things in themselves"? This includes plurality, how do we know if there is such a thing? — Manuel
For two reasons, common sense and textual.
Common sense
What kinds of objects are things in themselves.
From A29 of CPR: For in this case that which is originally itself only appearance, e.g., a rose, counts in an empirical sense as a thing in itself, which yet can appear different to every eye in regard to colour.
From A272: Of course, if I know a drop of water as a thing in itself according to all of its inner determinations, I cannot let any one drop count as different from another if the entire concept of the former is identical with that of the latter.
By common sense, does anyone not believe in the existence of roses, drops of water, houses, camels, trucks, etc. Would anyone crossing a busy road and seeing a truck bearing down on them actually think that the truck, the thing in itself, doesn't exist in reality but only as a figment of their imagination. Wouldn't everyone make sure they quickly got out of the way. Doesn't everyone believe in that things in themselves exist independently of their own thoughts of them.
Would Kant have not worked at the University of Königsberg for 15 years if he thought that the thing in itself, the University, only existed in his imagination and not in reality.
Can anyone argue from common sense that Kant was not a Realist.
He was clearly not a Direct Realist, as the Direct Realist believes they have direct and immediate knowledge of the thing in itself.
His philosophy, as @mww writes "Matter, as such, cannot have a name, which is a representation derived from the synthesis of conceptions, hence given from thought, not sensation" is that of Indirect Realism, the view of perception that subjects do not experience the external world as it really is, but perceive it through the lens of a conceptual framework.
Textual evidence for the existence of the thing in itself
Kant makes numerous statement against the charge of Idealism.
In the CPR Anticipations of Perception, sensation can be understood both as stemming from the object of experience as well as the thing in itself. This duality may only be understood if the transcendental is distinguished from the empirical. Kant's Theory of Affection allows for two different explanations, though together make a coherent whole account of human knowledge.
It would be a mistake to conclude that because a thing in itself remains indeterminate it cannot exist. Common sense tells us that there is something behind an appearance, even if we don't know what it is.
Kant wrote in A536 of the CPR that appearances must have grounds that are not appearances.
If ... appearances are not taken for more than they actually are; if they are viewed not as things in themselves, but merely as representations, connected according to empirical laws, they must themselves have grounds which are not appearances. The effects of such an intelligible cause appear, and accordingly can be determined through other appearances, but its causality is not so determined. While the effects are to be found in the series of empirical conditions, the intelligible cause, together with its causality, is outside the series. Thus the effect may be regarded as free in respect of its intelligible cause, and at the same time in respect of appearances as resulting from them according to the necessity of nature.
We know that objects of experience, although they are mere appearances, are given to us. If the appearance has not been generated by the knowing subject, then an external cause must exist, even if the external cause is unknowable. The thing in itself allows for the very possibility of appearance.
There are many passages in the Fourth Paralogism where the thing in itself is declared as the cause of appearance, and even in the Second Analogy, the thing in itself is described as the source of affection.
Summary
Kant is clearly a Realist, and in today's terms an Indirect Realist. Kant's Theory of Affection
may be read as not only that sensation stems from the object of appearance but also from the thing in itself, not a contradictory position, but two parts of a coherent whole.