Java Stream HOW TO .stream()

Breaking this down for you:

Set<String> allDatabases = new HashSet<>();
for (Entry<UUID, Expression> entry : alleMap.entrySet()) {
    allDatabases.add(entry.getValue().getChapter().getDatabaseName());
}

This is getting all unique database names for all chapters for expressions that are values in your map.

 List<String> allDatabasesList = new ArrayList<>(allDatabases);
 String[] result = new String[allDatabases.size()];
 for (int i=0; i < result.length; i++) {
    result[i] = allDatabasesList.get(i);
 }  

As far as I can tell this is all just to convert the set of database names to an array. If so, you could have just used allDatabases.toArray(new String[allDatabases.size()]. Streams have a similar method (though they take an IntFunction to create an array rather than the new array):

This translates to:

alleMap.values().stream()
    .map(Expression::getChapter)
    .map(Chapter::getDatabaseName)
    .distinct()
    .toArray(String[]::new);

Note the distinct to replicate the uniqueness of sets.