It's even worse: Even though Dutch and German separated only a few centuries ago, as a southern German I wouldn't have understood much of your Dutch example before I started to learn Dutch recently. The same problem exists with Yiddish, the other language that is similarly close to German.
And as if that were not enough, the same problem exists with dialects of one's native language. I mostly know this from German, but it's true for English as well. Most native English speakers will have quite a bit of trouble understanding even standard Scots, which is somewhere on the boundary between a dialect of English and a separate language. But thick dialects of English proper such as Yorkshire English can be even harder to understand. (Shakespeare's original pronunciation is relatively easy in comparison, but it's still relevant because it demonstrates how pronunciation changes over time. Compare Richard III's pronunciation just a little further back, or even Chaucer. As you can see, pronunciation has changed much more than spelling, which is one reason why modern English spelling is so chaotic.)
As the examples show, the problem is in the different pronunciation. But once you get immersed into the culture surrounding a dialect of your native language, or a foreign language close to your own, there are two things that make it very easy to pick it up:
- There are regular transformations connecting the pronunciations. Once you have realised that bee-uck is Yorkshire accent for book, you will form a theory that look, hook, cook will be pronounced lee-uck, hee-uck, kee-uck. (I don't know whether any of the last three examples is correct, and I wouldn't know where to check. In general these theories are correct in more cases than they are wrong, and that's enough.)
- Even where words differ, such as ken for know in Scots and Scottish English, dialect words are often either related to words in the standard language or even exist there with roughly the same meaning, though in the obscurer parts of an educated speaker's vocabulary.
My personal experience has been that as a native German speaker, through very little effort and in just a few months, I passed from understanding hardly any Dutch at all (written or spoken) to understanding it pretty well though still basically unable to speak or write it. The key to this was:
- learning a very small number of important differences of grammar;
- learning a very small number of important words that are totally different or false friends;
- internalising the different orthography and the mostly regular pronuncation shifts (this happened automatically along with the previous two points).
With that I was able to read Dutch books quickly with very little trouble and no dictionary.