To start with, bear in mind that Shirou Emiya is a 16- year old teenager, and thus, prone to doing anything a teenager does at that age. Now, add an almost suicidal tendency to protect everyone he meets, alongside a broken sense of self from a past incident, and you'll pretty much get Shirou's base character. While he is easy to get along with, mostly everyone might get annoyed at how he is pretty much someone who does whatever he can to help people, which includes sacrificing his own life without thinking twice.

The reason for his current self was simple: The Great Fuyuki Fire. Being one of the survivors of the incident, Shirou's child self was broken by the sight of dead bodies around him, so in a sense, you could say that the fire killed the person Shirou used to be. It was not until a man saved him that his old self was reborn into Shirou Emiya, a male who wished to be a hero, to emulate and walk through the same path that the man walked on, saving everyone he meets along the way.

The other reason would be his survivor guilt, which was born from the fact that he actually survived a disaster that claimed thousands of lives, which he saw as he walked through the fire. The sight of people dying around him without being saved pushed him to find a way to atone for his inability to help them while he was still a kid, and one such way to atone was to die in his own terms, preferably while saving a life rather than taking them. This broken shell was not fit to live, so it's only natural for him to pursue a way to die saving someone rather than a normal one.

As a teenager, Shirou is definitely a social person, who can befriend practically anyone he meets, which is displayed by his... strained friendship with Shinji Matou, who is basically the jock of the school. Not only that, Shirou will always attempt to do something to be of help to his friends, be it something as mundane as fixing things, to even more elaborate things such as setting up a neighboring class' stall during a school festival of some sort. Naturally accusations of being too exploitable are abundant which is not surprising considering his habits.

To explore this even further, his desire to save people leads him to do what any protagonist does when any of his friends are in any form of danger: Jump into it. He had done this so many times over the course of his story that people called him out on being dangerously selfless to the point it does him more harm than good. Again, this all started from the fire he survived, which, in addition to killing his child self, caused him to have a survivor's guilt over all the lives lost during that very same event, and thus, to see fit to devote his life in following part of Kiritsugu's path in order to save everyone.

Often, this was what caused him to butt heads with Archer throughout the whole route, Archer was a pragmatic person who condones such acts as leaving Caster alone to gather mana, and possibly off the other Servants versus defeating her there and then in order to prevent a lot more people from dying. And then there are the other occasions where Archer actually tried to kill Shirou, causing seeds of distrust, and possibly disgust to form within him, marring his opinion of the definitely older male. His betrayal of Rin, in fact, further stamped this down to his heart, and there's no doubt in the fact that their relationship in the game will be really volatile.