Well the same question is frequently asked me my most of my friends here in Italy!
I believe that people re/enact for all different sorts of reasons, and it is not really easy to find one common motivation behind all re/enactors.
For what it is worth I can just mention my own personal experience .
I have come to the re/enacting and Period Costume Shooting world as a result of a sort gradual evolution of my hobby as a history buff .
Since a child (I am now 49) I have been fascinated by military history. As a result of this I started on a long career of military modelling , continued, to a lesser degree now, to this day.
Being a military modeller means I needed to get the right reference material and information in order to make accurate models, which means a buying a growing number of books and frequent visit to museums.
The result of all this researching activities was that slowly but steadily i found myself shifting my aim from the scale model world to the real things , and so I starting collecting original books, militaria, and weapons relating to my favourite periods.
Eventually , I gradually started to wish to move to a sort of 1/1 scale modelling in which by using your own body and a mixture of original and reproduced items you can get an as close as possible to appearance of the subject/s you like most. T
Once I got my outfit right than I started asking myself what I could do dressed like that!!
So I inevitably got in touch with other re/enactors , and re/enacting events provided the right context where I could wear the uniform I liked most, without looking like someone who had just missed a fancy dress party !
I soon made new friends in the re/nacting world and now and more often then not taking parts to events and organizing them has become almost as a perfect excuse to spend some of my free in company of friends sharing the same hobbies and interests, providing at the sane time , in good weather at least, a good opportunity for some outdoors activities.
The next, almost inevitable step, was Period Costume Shooting.
While for obvious reasons (in Italy at least) re/enacting is mostly with fake or deactivated fierearms, gradually , like most re/enactors I sarted to develop an interest in the real proper functioning firearms , and therefore the wish to own and actually shoot some of the original guns from my favourite historical period.
For me this has always be the British Colonial Period, The Old West, WWI and WWII , and the Viet Nam War.
Reading about Battles of all these periods is a definitely necessary background , but to actually experience the smell of the cordite (or black powder as appropriate), and feeling the differently kicking of the butt against your shoulder when you fire a Springfield .45/70 carbine, a .303 Lee/Enfield , an M1 Garand or an M16A1 , etc, properly dressed in a hisctorically accurate period uniform , surely helps some peple like me to feel even closer to the period they like. And even the apparently less pleasant task of cleaning an oiling ones weapons after a shooting day , also becomes part of the game, contributing to a better understanding of firearms functioning and required maintenace operations that troops in the field had to carry out .
During the last couple of years, in Italy Period Costume Shooting has become a sort of hobby and sport in its own terms and we have now an increasing number of competitions in which competitors attend , say, dressed as a German Afrikakorps Officer when firing a Luger P08 and Mauser K98, or in Desert Rat Outfit with SMLEs and Webleys.
Here the accuracy of uniforms is obviously essential for both participants and spectators alike, adding to the show, as we would say in Italian!
Here in Italy we are now planning to develop a pre/WWI and Victorian Era Costume Shooting group. At the start I will be probably the only British Colonial (I am putting together a First Sudan Campaign British Infantry Officers uniform), since most would be Italian Risorgimento Period, but we do not dispair to add more Victorian Colonials with time including hopefully some convincingly looking Zulus and Dervishes!
So , to get back to the original question ,at least in my case , re/enacting was a gradual evolution > it all started with building an Airfix Bengal Lancer and a box German Afrikakorps infantry , and in a process lasting more than 30 years it has ended up in shooting Lee/Enfields and Lugers in full uniform!!!
"What a pity you are not an Englishman!".
Gordon's remark to his Italian lieutenant, Romolo Gessi