The fighting liberties of a player in a semeai is a specifically defined number of relevant approach liberties so that they are equal in the unsettled, non-trivial cases of a semeai class and type. 
In a basic semeai of the type "one eye versus no eye", the inside liberties belong to only the group having the eye. However, all the inside liberties are physical liberties of both semeai strings and also approach liberties of both semeai strings. One cannot say "The inside physical liberties belong to only the black semeai string." and one also cannot say "The inside approach liberties belong to only the black semeai string.". To express the intended meaning properly, another kind of liberties is used and one may say: "Only the black string's fighting liberties include the inside liberties."
The white semeai string has 9 fighting liberties; all are on its outside.
The equal number of fighting liberties indicates that the semeai is unsettled. Their difference 9 - 9 = 0 gives us this essential information.
For reference, the approach liberties (here they coincide with the physical liberties) are shown in the following diagrams. They do not inform well though about the semeai status or relevant liberties difference.
These liberties Black would need to occupy to remove the white semeai string.
The difference 18 - 9 = 9 of approach liberties does not tell us the semeai status.
The concept's origin is unknown. Richard Hunter in Counting Liberties And Winning Capturing Races used the concept in his book but at the same time used liberty for the three different concepts physical liberty, approach liberty and fighting liberty. Robert Jasiek invented the latter's name to dissolve the ambiguity and to introduce clarity in the terminology and basic semeai theory.