The function next_node returns the license number of the next node (including its descendants), as well as the rest of the input (remaining numbers). In line 20, we invoke this function with the entire input, and so here we expect the function to consume the entire input (i.e. there should be no remaining numbers). The purpose of this pattern match is to take the license number and also to assert that the function has indeed consumed the entire input. If any number remains, either the input is invalid, or there is some bug in the code. In both cases, we don’t want the program to proceed further.

Note that this pattern match is the same as {license_number, []} = next_node(numbers, license_strategy), i.e. without the _remaining_numbers part.
The _remaining_numbers = addition is included only to clarify the meaning of the [] element.

This works because pattern matches can be chained. You can e.g. write something like

entire_result = {license_number, remaining_numbers} = next_node(...)

In this case you take the entire result tuple in entire_result, and also take each element of the tuple into license_number and remaining_numbers respectively.

Another piece of the puzzle is that if you don’t need to use a variable, you can prefix it with underscore. So therefore the _remaining_numbers = [] is a chained match which states that the second element represents remaining numbers, and that it has to be an empty list.