These enclaves/exclaves are located along the border of the European countries of Belgium and Germany, just south of the city of Aachen. Here is an example of what these borders actually mean at some locations:
Pretty surreal, isn’t it? If you would like to know more about why these strange borders were created, read their history below.
What are enclaves and exclaves?
First of all, let’s clarify what exactly enclaves and exclaves are as well as the difference between them.
The explanation from Wikipedia describes them quite well:
An enclave is a territory that is completely surrounded by the territory of one other state.
An exclave is a part of a state or territory geographically separated from the main part by surrounding alien territory.
So following the definitions, these German territories are exclaves of Germany and enclaves of Belgium.
Why do these particular exclaves exist?
The reason for their existence is one to find in the history books.
The area was annexed by the Kingdom of Prussia in 1822 and it became part of the Rhine Province. 
An Act in 1882 stipulated the construction of a railway to help integrate the borderland better into the newly unified German state. The Germans opened the line from Aachen to Monschau in 1885.
Soon it was expanded south, reaching the town of Ulflingen (today Troisvierges, Luxembourg). This link was quite important because of industrial interests. Major iron ore deposits were discovered in Luxembourg and Lorraine at the time and the need for a transport corridor to them was inevitable. 
The transport of coal began southbound from Germany to fuel the booming steel industry in the region, which in turn supplied the raw-material needs of the Ruhr. The entire line soon became known as the Vennbahn.
After the war
The German defeat in World War I and the provisions of the Treaty of Versailles completely transformed the political map of the region. The treaty moved the border eastward by as much as 20 kilometres, and most of these former German territories became part of Belgium.
However, the question of the Vennbahn was quite complex. Even though some of the lands west of the line were to remain part of Germany, Belgium claimed sovereignty over the trackbed and the stations. The Belgians argued that the railway was a vital communication route for their new eastern territory given to them by the treaty.
The commission which was set up to decide the matter agreed and the Raeren-Kalterherberg section of the Vennbahn was ceded to Belgium in 1921.  It took a further year to finalize the details that left five German exclaves – called Munsterbildchen, Rötgener Wald, Rückschlag, Mützenich and Ruitzhof – west of the railway.
The line remained in service until 2001 before it was dismantled in 2007-08.
The five enclaves today
(from north to south) 
Munsterbildchen Area: 182 hectares. Population: 50.