I never met a Dutch who doesn’t understand German … (Of course that's just episodic and not representative. )
Probably because they got German television (though I can’t understand why they watch it …). Once when I was in Groningen - about 70 km from the German border - we saw people watching the German “Sportschau” in many houses, a television sports magazine. O_o (They have no curtains in the Netherlands …)
But the two languages are also quite similar. I as (Northern) German understand about three quarters of written Dutch texts, of course not so much when they talk, but that depends on the region where the Dutch come from. When it comes to dialect, the Eastern Dutch Nedersaksisch is … just Niedersächsisch, the Low Saxon dialect/language (though massively influenced by Dutch), we just call it Plattdüütsch (literally Low German) in Germany.
I found a map about the Northern German (mainly Low Saxon)/Dutch (mainly Low Franconian) dialect continuum
Yellow/brown means Low Saxon dialects (Plattdüütsch), reddish means Low Franconian dialects (Dutch); the grey parts in the north of the Netherlands and Germany are Frisian language areas. Ostfriesisch (Eastern Frisian) is not a Frisian language but Low Saxon on a Frisian substrate (some words, grammar, pronunciation); 15 is Niederrheinisch, just typed wrongly …
Just to show the similarities/differences between the languages, a few sentences in relevant dialects:
Dutch: Er zijn rapporten die een dramatische afname van streektalen ten gunste van het Nederlands aantonen.
Oostfreesk Platt (“Eastern Frisian Low Saxon” O_o): Dat givt Berichten de up een dramaatske afnahm van streektalen togunst van dat Nederlandske henwiesen.*
Plattdüütsch (Low German/Saxon): Dat gif(f)t Berichten de op een dramaatsche afnahm vun regionaalen Spraken togunst vun dat Ned(d)erlannsche henwiesen.*
Standard German (literal translation): Es sind Rapporte, die eine dramatische Abnahme von den (Land-)Strichsprachen zugunsten des Niederländischen anzeigen.
(Nobody would say it this way … It’s just Google German - a literal translation.)
Standard German (like it would really be used): Es gibt Berichte, die auf einen dramatischen Rückgang der regionalen Mundarten zugunsten des Niederländischen hinweisen.
English: There are hints that indicate a dramatic decline of (regional) dialects in favor of Dutch.
* means there are more varieties: these dialects differ slightly from village to village; aantoenen is also a Low German word, but we won’t use it in this context.
As one can see, these languages have the same words but not always the same use of them, and the differences mostly lie in (at least regarding the spelling) different realizations of the same morphemes (af / of, op, up / ab; ten / to / zu). Dutch and German “v” are (mostly) spoken like “f”, not like the English “v”.
Because of the second consonant shift in Germany, Low German is, just refering to the sound, by far more similar to Dutch than to Standard German. If you know about the second (High German) consonant shift in Germany, you can easily transform Dutch and Low German words into Standard German. - The English translation shows the massive influences by Latin/French.
One more:
Standard Dutch: Nederduits is de verzamelnaam voor een aantal in het noorden van Duitsland en het noorden en oosten van Nederland gesproken West-Germaanse dialecten die niet hebben deelgenomen aan de Hoogduitse klankverschuiving.
Low German (Oostfreesk): Ned(d)erdüütsch (Nederdüütsk) is de Sammelnaam för een Antahl van in’t Noorden van Düütschland (Düütskland) un in’t Noorden un Oosten van Nederland sproken west-germaansche (germaanske) Dialekten, de nich (neet) deelnommen hebbt (hebben) an (aan) de Hoogdüütsche (Hoogdüütske) Luudverschuvung (-verskuving). (In brackets one Eastern Frisian variety)
Standard German: Niederdeutsch ist der Sammelname* für eine Anzahl von im Norden Deutschlands und im Norden und Osten der Niederlande gesprochenen west-germanischen Dialekten, die nicht an der Hochdeutschen Lautverschiebung teilgenommen haben.
*one would usually say: die Sammelbezeichnung
