For a large part it's down to the reproductive and match making success of two people.
Queen Victoria was born in to the German house of Hanover and she married Prince Albert of the German House of Saxe-Coburg-Saalfeld (it later became Saxe-Coburg and Gotha) they had nine children all of whom married in to other European Royal Houses.
Christian IX of Denmark was born in to the German House of Schleswig-Holstein-Sonderburg-Glücksburg (a junior branch of the German House of Oldenberg) and he married Louise of the German House Hessen-Kassel. They had six children who married in to European Royal Houses.
Queen Victoria and King Christian IX are there for known as 'the grandmother' and 'the father in law' of Europe respectively. Between them they produced 15 Germans who married other European Royals.
Nicholas I of Montenegro is also sometimes known as 'the father in law of Europe' since his five daughters were married in to other European Royal families. He's was not German, but he's worth mentioning to stress that not every European royal line is entirely German!
The reasons that Britain and Denmark had German Royal families in the first place are as follows:
In Britain once the Monarchy was restored after the civil war Parliament wanted to ensure that there wouldn't be another Catholic Monarch. They passed a law barring Catholics from inheriting the throne. When Queen Anne died childless her nearest Protestant relative was the German Georg Ludwig of Hanover who became George I of England.
In the case of Denmark they had been ruled by the German House of Oldenburg since the 15th c. (It was an elective Monarchy until the 17th c. with the right to choose the Monarch being the privilege of the Privy council.) Count Christian of Oldenberg was chosen during a spot of turmoil because of his maternal connections to the Danish Royal line.
I think the whole concept of elective Monarchy can surprise people because they automatically assume that Monarchy has to be strictly heredity, so here's some more on that http://en.wikipedia.org/wiki/Elective_monarchy
England had an elective Monarchy until the Norman conquest, the usual practice for Germanic lines (which Anglo-Saxon England was) was that the line of male primogeniture (usually the eldest son of the last King) would be followed but the electors had the right to appoint another member of the Royal family if this line led to someone deemed unsuitable.
There's another really big reason. Look at a map of Europe from the 1600s or 1700s. Count up all the different individual "nations". After 1603, the British Isles just had one crown, one royal family. So were France and Spain (and, heck, for a while, Portugal wasn't even independent/didn't really have a royal family then).
Now. Take that map, and pan eastward. You see that big complicated multi-coloured patchwork quilt right in the middle there? That is what is now Germany. In the 17th and 18th centuries, depending on how you define "sovereign" or "independent", after the 1648 Peace of Westphalia, there were something like 300-400 different little countries in what is now Germany. And each had their own royal family; of course, there'd be a lot of interconnections between the royal families of the various German states, but still, eh?
Yes, Queen Victoria's fecundity and mathcmaking prowess had a lot to do with it, but even before Victoria ascended the throne — hell, before any Hanoverian ascended the British throne (George I being the first, of course, in 1714), the sheer multitude of German monarchies meant that (to avoid the kind of inbreeding that the Spanish royal family ended up with) they had to farm out young princes and princesses to marry into different families.
Two questions for the pro historians here: First, is there anywhere that has a good reliable list of all the little German states, before and after Westphalia? Second, has anyone put together any good comprehensive royal/noble genealogical databases of that period (call it 1648 Peace of Westphalia to 1815 Congress of Vienna)? It'd be very interesting to try to trace all the interconnections.