In most stories women often play the role of the damsel in distress. The character that always needs saving and is often the prize for the hero. Because of this role being used ever so often in the majority of stories women are often stereotyped as being weak, timid and often used more a sex symbols rather than being more complex and deep characters. Cinema only increased this since most female characters were very attractive actors that suited the male gaze (shows that cinema is more focused on the male audience compared to female). Now this doesn't apply to every role a women plays since there females villains who are vindictive and evil like: poison ivy, Jason's mother etc, but also some female characters are quiet strong smart and poses the characteristics that are often seen within the masculine hero like Ellen Ripley, Sarah Connor, Samus Aran etc. This type of character is most present within horror films (slasher in particular) since this character is the only one smart, strong or brave enough to escape/defeat the killer where as everyone else (including male characters) have been killed. This is known as the 'Final Girl'.
The final girl theory first became popular with such horror films as alien, Halloween, and Texas chainsaw massacre in which the only characters to make it to the end and live where females. Throughout the course of these films the other characters are picked off one-by-one by the killer, often due to the characters doing something stupid or not being strong enough to confront the killer and once the film reaches its climax only one person is left and she is the one who ultimate defeats the killer or survives (e.g. fight back in Halloween, escaping in Texas chainsaw massacre, and killing the killer in alien). Because of these type of films females have been portrayed as much stronger and more independent characters compared to their traditional role of the damsel in distress, and this often makes the film more exciting to watch since it's something different compared the traditional masculine hero.
Often what makes the final girl stand out isn't the character herself in particular but rather the killer and the side/stock characters. This is mainly because the stock characters in most horror/slasher films are generic and one dimensional character that are built around typical teenagers stereotypes (since most character within a slasher film are teenagers) like a nerd, a jock, a whore etc. These characters serve no other purpose other than providing a cheap laugh and are the murder victims of the killer which is why at the end of the film we want the final girl to survive and overcome the killer since she is the only character we care about (though this isn't present within every horror film since the characters in alien are all interesting and it's sad when they die). On the other hand there is the principle of the killer whom we often follow throughout the first and second half of the film as he kills off each character from his point of view, so we feel like we're him and know how dangerous he is. Once the third act takes place we take the view point of the final girl and we want her to survive since we know what the killer is like and how dangerous he is and because we can see him properly from her point of view this makes him seems scarier since it looks like he going to kill us instead of us being the killer.
Because of these factor the final girl is always the most developed character since she has the most screen time and the audience connects with her more than any other characters which is why when she overcomes the killer there is a great feeling of satisfaction as well as the fact that the character is female rather than male which is a welcome change. What's interesting is that the final girl theory could also represent some sort of real life issue since with most of these films the killer is a man and the final girl is female, this principle/concept could represent overcoming male oppression since males are often seen as strong more dominant beings than female and this concept could be representing those who believing against this idea and even have experienced this in their lives. The character of the final girl also plays against the expectations of the male audience since not only is she head strong, brave and even possibly masculine, but also this type of character from other females and sometimes doesn't meet the male gaze compared to the other females within the film (though that's mostly down to opinion). However in some cases the final girl does sit nicely between these expectations since she is strong, smart and interesting enough to watch and care about but not to the extent in which she becomes overly masculine that she doesn't meet the male gaze (e.g. Ripley from aliens).
The Final girl has also been a very large inspiration for strong females in various types of media since the majority of strong females leads are strong, smart, interesting and also meet the male gaze which is what makes them such likeable and recognisable characters since they have both masculine and feminine characteristics and often are very attractive (e.g. wonder woman, rogue, Samus Aran, tomb raider, Chun-Li etc). In the end the final girl theory is simply the identification of certain characters with in a female character in horror films that often play against typical roles females with in storytelling and helps to establish women as much stronger, independent people compared to their stereotypical role of damsel in distress. It's because of these characteristics that the girl is able to outsmart the killer and live at the end due this character being the most developed and cared for by the audience which is often helped by the generic stock characters and taking the view point of the killer so that we can understand how dangerous he is and why there is so much fear when the final girl has to confront him and so much joy when she comes out on top.