Ok, I'll have a go at this. If someone with a more detailed knowledge comes along I'm sure they will correct me.

The German Empire after unification in 1871 was a federation. We'll get into how it worked in a minute, but first it is worth noting that federalism in one form or another had actually been a common system in German history: for example, the authors of the Federalist Papers at the time of the US founding referred to the Holy Roman Empire, which included most of the German States, as an example of a federal (or at least porto-federal) system. The various kingdoms, principalities, bishoprics and city-republics enjoyed substantial autonomy under the overall - and mostly indifferent and indirect - rule of the emperor.

Napoleon put an end to all that, but after Napoleon's fall the German states were again (loosely and imperfectly) confederated under the Bundesakte of 1815 source.

The point of all this background is that the German states before 1871 were used to firstly, exercising considerable autonomy, and secondly, having that autonomy constrained - however nominally - by a higher level authority. Each state already had its own ruler and government, its own courts and legal system, and in most cases by the mid-19th century, its own parliament and constitution. In other words, they had significant governance capacity. The German Empire didn't need to rule the whole of Germany, in terms of making laws, carrying out administration, and levying taxes. The German states could do that through their own institutions. The Empire only had to rule the states. It could acquire states by means of negotiation and could co-opt existing state institutions to serve imperial purposes.

Because it had to take these existing states into account, the German Empire developed a very unique (for its time, now more widely copied) type of federation, in which the Empire made use of the administrative capacity of the states. There were Imperial Laws which were administered by the states (in contrast to, say, the US system, where in general terms state governments enforce state law and the federal government enforces federal law). In principle, this gave the states some latitude in how they would apply Imperial law (I know more about comparative constitutional history than German history, so I can speak only about how the system was supposed to work - I don't know the extent to which the states actually made use of this power). The states would also have a direct voice in the formation of law at the imperial level, through the Bundesrat - a second legislative chamber that represented the state governments.

It's worth mentioning here that German federalism (1871-1918) was very unbalanced, because Prussia held a predominance of power, including a very substantial voting bloc in the Bundesrat that gave it an effective veto. The Emperor was also double-hatted as King of Prussia, and the Prussian government, with its big bloc of Bundesrat votes, was responsible to him (this feature was one of the things that made the German Empire, despite the existence of a national parliament elected by universal suffrage, less than fully democratic). Prussia remained the dominant entity throughout this period (it was dismembered only after the Second World War).

Main source is Daniel Ziblatt: Structuring the State: Formation of Italy and Germany and the Puzzle of Federalism

Hamilton, Madison, Jay: The Federalist Papers

Friedrich Heer: The Holy Roman Empire