As DevSolar noted in his answer, churches generally are tall, their spires even higher, but to answer

Q Why in most German places is the church the tallest building?

it might suffice to say just: No. Church spires are now usually not the tallest structures in most German places. That is an assumption quite far from visible experience on location or from written list of records. But as broad and a founded on an unproven assumption the question is, for what is there, we have to observe a multitude of influences, if not reasons.

For a while though, they often were. As community projects, at least co-financed by the church, they served a much different function than residential or commercial buildings. Touching the sky as being near 'the realm of God', signaling wealth, prestige and devotion, all offset the inherent disadvantages of building high: unstable structure, more empirical than statically pre-caclculated construction, price, and usability of enclosed space.

The "building" definition is indeed a curious one.

For that we need a bit more detail.

In German there are three categories to distinguish:

1. Gebäude, 'building' in English, but as the pictures in Wikpiedia show, most people would include churches within this category.

   "Buildings are independently usable, covered structures that can be entered by humans and are suitable or intended for the protection of humans, animals or property.

2. Bauwerk, 'edifice, (building) structure'. Pretty ill-defined, larger, more inclusive category, no proper English translation displaying the differences.
3. Turm, 'tower'.

   A tower is a walk-in vertically aligned structure that is defined by its height. This means that its height is either a multiple of its diameter or its thickness and/or it clearly exceeds the surrounding buildings or adjacent components (e.g. the nave of a church or the wall of a city fortification). A tower can stand on its own (real tower, e.g. round tower), or be part of a larger building or structure (e.g. church tower, minaret; towers as part of a castle or city fortification).

These are all not defined precisely and without overlap on Wikipedia. The civil service and legal definitions may be a bit clearer. But colloquial speech is even fuzzier than Wikipedia.

The church steeple with its spire is therefore often just a part of the 'real' building and counts for its height either on its own, or has to be counted separately. Whether counted separately or pars pro toto for the entire church building's height, that means one should either look simply at the highest man made structure including all kinds of towers, or one has to look at the unquestionable building-characteristic of a church and look at the roof height over the nave.

If we go by artificial structure, we also get Colonius Tower (Cologne), Olympic Tower (Munich) etc. The main difference observed is probably churches do not have more floors in use than residential buildings of their time (mainly ground floor, large open space, emporium, 1–3 galleries, the rest mainly service personnel reserved?). Radio towers do not service with God, but serve a similar function in communications? I'd say that if some people feel the need for it, and have the will, they build stone circles, Frauenkirchen, radio towers or pyramids.

From the definitions we have to conclude that the proper and unquestionable 'building' part of a church would have to be measured as just up to its roof height over the nave. That means that the highest church spire in the world in Ulm would be impressive on its own with its 161m. But the nave is only 41 m high. This phenomenon applies of course to most churches.

For Ulm this further means the church-building does not make it for the top ten of 'proper' buildings, the highest measured in as 82 m. But if going just for the highest structure in Ulm then the radio-tower measures 162m and an industrial 'edifice' comes in 125m.

For Eastern Germany we can also observe the official socialist city planning until the 1970s explicitly called for erecting 'city dominants', large buildings the specifically should be built 'higher than church spires.'

For the little Jena that means the important St Michael church clocks in 50m for the steeple/tower, but the JenTower, with fully occupied floors comes with 144m.

This doesn't mean that this 'dominance' was a somehow 'communist idea'. For Dresden we observe the radio tower to be 252m tall. The more famous but only second-tallest church, Frauenkirche church, built in the 18th century measures 91m. But two industrial towers are higher than the church. More important concerning 'dominance': The tower on city hall was built from 1901–10, and the inhabitants also explicitly called for it to demonstrate the 'citizens power' and so they built it 100m high.

Taken together, the present situation displays that most cities, towns and villages now have as highest man-made structures, colloquially called 'buildings', communications towers, industrial edifices, wind power stations. Only in small towns then come church towers, while in larger towns and cities we see commercial offices, residential high-rises being taller since around the 1960s. Just a minority of old settlements that weren't destroyed in recent wars or by capitalist development strategies –and thus often have an intact 'old-town' – still look as if the church is the highest building. 'Look' because most grew and when sprawling into the country-side gathered high-rises and towers as well.

Since before the elevator, climbing 5 stories was mandated to be done on foot. And this is still quite a drag for most people, or for older folks even bordering on the impossible. When the churches were the highest structures, residential housing rarely exceeded 5 floors. And within churches the same effect is observed. In houses the ground or better yet first floor were the beletage, and within churches the room height may be impressive, but the usable floor space much reduced in comparison. Only the spires of churches went up for really no practical reason.

As such, today in most German, if not European cities, houses still aren't the biggest buildings, or man-made structures. Now the industrial chimneys, radio towers or wind craft engine towers top it off over church spires and regular skyscrapers.

Add to that the effects that in several places you just cannot go any higher: in several cities there were and are building regulations in place, some had to do with health and safety, some with mere optics and a few indeed that the church had to remain the largest visible structure. But these are rare in comparison to a lot of physical constraints the environment presented in many places. Like in Paris, many geological situations make it impossible to put too much pressure in the form of mountainous stone structures onto the ground. Pisa's tower is evidence how lucky they got until now with this 'unfounded aspiration'.

But that's really getting us back to the "why" for the relatively short time that church spires were the tallest structures in town in Germany:

This is a reconstruction of Bologna in the 11th century! An unsustainable development with no apparent practical use. This skyline is no longer there. But a few remnants of this style are visible in today's San Gimignao

These are all medieval structures, and the church is clearly not the tallest of them. This style was copied in Germany, among the affluent trader families. In Italy, the communities came to the decision that these were unsafe and wasteful showoffs, so they had to be dismantled down to few floors, if they didn't come crashing down in earth quakes already by then.

In German Regensburg for example we still see The Golden Tower

next to the cathedral.

Problem here was, the private tower is 9 floors and 90m tall, built in 1250 and completed in a fairly short time. The cathedral was only started in 1275 and took until 1450 to be usable; not: finished. As so often, the spires then weren't even as high as they are today. Around 1860 they looked like this:

As was noted in comment below the question, the claim rests on the assumed instruction: "no building taller than the tallest church". That this cannot be the sole governing rule after ~1520 seems evident if looking at the confessions: Why should a Protestant house be allowed to be bigger than a Catholic church, if in a Catholic region the Protestants happened to be a bit richer so they could have built a slightly taller spire to their congregational structure? Even if this seems to be entirely counterfactual, we might see the potential for ample conflict out of jeopardy for such an ecumenical guideline.

Summary

For most of the time there was no practical need for anything that high, almost no practical means for anything being that high. Costs were often prohibitive, the usability very limited. Only for a very short time there were community restrictions as the only thing limiting structural heights. Since they either fell completely or exceptions on application were increasingly granted, the special place for church spires as highest points on man-made structures was gone again. It was always a combination of conditions, of hard facts stemming from material realities and community decisions. In reality, whenever a need was seen, the will as well as the finances were existing, then people build as high as they could while buildings still remained usable.