((Note: "WNT" = "Woordenboek der Nederlandse Taal", "Dictionary of the Dutch Language". Links are provided as a source, but it's all in Dutch.))
The word Duytsch (of which Duytschen is an inflection) etymologically is the same as Duitsch and Dietsch (WNT). Back in the day there was no official spelling and Duitsch/Dietsch was mostly a regional variation, with Dietsch being the oldest form (WNT).
Dietsch was a derivation the word diet, a Middle Dutch word which meant 'people' (WNT). Diet fell out of use at the end of the Middle Ages, but Dietsch (and variants) remained in use in two meanings:
In the 16th century people gradually started to make a distinction between Duitsch/Duitschland (German/Germany) and Nederlandsch/de Nederlanden (Dutch/The Netherlands) (WNT mentions the earliest use in the 15th century). This eventually became the clear-cut distinction it is today, but language change occurs slowly and for quite some time (certainly during the 16th and 17th century) Duitsch could be used as referring the Dutch instead of German.
This is why you get people arguing - to this day - whether the Duytschen in the national anthem refers to Willem of Orange being Dutch or German, or just as 'of the people'. Any opinion on this matter can be defended.
That said, I am convinced it was used in the meaning of 'German'. I realise this is not a very patriotic interpretation and many of my countrymen will disagree, but I find one thing very convincing:
In 1580 William of Orange published an Apology, which was a reaction to his banishment (including 25000 crown bounty) by Philip II of Spain. In this Apology, William defended his actions and accused Philip and his advisors of grave misdeeds. William also comments on his own background. Notably, he says: "ende wy sijn gheboren in Duytschlandt, met d' welcke dese landen natuerelicke vrientschap ende eenicheit onderhouden". ("We are born in Germany [Duytschlandt], with which these countries [the United Provinces] have a natural friendship and unity.")
Of course in the same Apology one can find support for the interpretation that William of Orange belongs to the people: "Hoe wel nochtans dat ulieden genoegh bekent is, dat wy van onsen ouderdom van elf ofte twelf jaeren af midden onder ulieden, ende niet elders, sijn opghetrocken, in der voegen dat wy dese landen voor sulcks vry moghen houden, alsof wy daer in gheboren waren." ("However it is well-known among you, that we were raised in your midst from the age of eleven or twelve, such that we may almost consider these lands as our birthplace.") [Translations made by me on the spot.]
Long story short: we're really not sure and we all have our personal opinion on the matter (my opinion being the correct one, of course).
The exact meaning of this part has been argued back and forth - it could mean of German blood, but another argument is that the word stems from the old High German word diutisc which means us, our nation or our tribe. This word is the origin for Dutch. The meaning would then be that William of Orange was not a foreign ruler, but a man of the same blood as the people.
Another explanation could be that the text refers to Dietsland (Dutchland) or the Greater Netherlands, which would include the Netherlands, Friesland and Flanders as well as Dutch-speaking areas of northwestern Germany - William of Orange would then be of Dietsen blood, a reference to the right of the Netherlands to rule all of Dietsland.
Personally, I find the second explenation, that William was of the same blood as the people of the Netherlands as the most plausible interpretation of the time - after all, the Dutch fought the 80 years' war to rid themselves of their Habsburg (Austrian and Spanish) overlords.