I haven't read any empirical studies myself, but Wikipedia refers to three resources that seem to support this claim, so you might want to consult those studies if you are interested in the details.
From a theoretical point of view, I find it not implausibel that capitalisation, as long as it happens in a systematic way, improves readability in introducing a graphemic distinction of certain syntactic elements.
As I wrote in response to a previous question, in German, at least compared to English (there might be more extreme examples), word order and syntactic relations are sometimes not as easy to determine especially in subordinate clauses, because German syntax doesn't strictly require SVO there. In speech this can be disambiguated by the use of intonation, which is obvisouly not available in written language, so you need to make use of other features such as punctuation or capitalisation.
Moreover, German's derivational system allows for relatively unrestricted nominalisation of other word categories, like die schöne (adj) Frau -> die Schöne (noun) or reiten (verb) -> das Reiten (noun), and as you can see, the derived words are in their phonetic form sometimes undistinguishable from their roots, so capitalisation might help emphasising the difference in their syntactic status.
Ahtough this is of course not scientific evidence, I'd like to add that especially in chat messages, where capitalisation is frequently omitted, I find it often harder to parse the sentence at the first try - although, of course, other factors such as inconsistent or missing punctuation and more tolerance for typos also play a role.
I'd also like to add that one should be careful about the results of such readability studies, because it might be that the difference in ease of reading is mainly or to some part due to the fact that speakers are simply used to the capitalised orthography convention, and struggle with non-capitalised nouns because they are unexpected. In fact, unexpectedness, manifesting itself amongst others in eye movement which one of the studies mentioned made use of, is what psycholinguistic studies frequently exploit to provide evidence about whatever linguistically motivated observations.
It might (!) be different if readers were used to non-capitalisated writing and the question would then be if capitalisation improved readability on the log-run despite it being unexpected first - but this is of course hard to test because those effects depend on the properties of the individual language so you can neither base those reults on evidence by speakers of languages which have by default no nominal capitalisation nor can you find speakers that are used to a different writing system of the same language.
Not claiming the studies did wrong, as I said, I haven't read them myself, you'd need to go into this more deeply on your own, just pointing out that measuring readability effects is not as easy as it might first seem.
The downside of such a further orthographical distinction is obviously that it makes spelling rules more complex, and there are people arguing for for the abolition of noun capitalisation in German.
However, I think that the largest recent orthography reform of German has eased this issue (and many other issues in my view, despite this apparently being an unpopular opinion) in systematising capitalisation to a point where I would claim that everything which would syntactically be analysed as something nominal is consistently written capitalised with almost no exceptions, so that capitalisation or non-capitalisation could indeed serve as a helpful distinction to improve readability.