Oh my, your question is a bit confusing. I am trying to figure out if you are asking why George shot Lennie, or if you are asking why the different interpretations of the ending exist. I will deal with both issues so that you are sure to get a firm answer. In short, I believe that George shot Lennie out of loyalty and friendship, and the different interpretations of the ending exist because Steinbeck meant for the ending to be ambiguous.
In regard to "why George shot Lennie," the answer is friendship and loyalty. Lennie has killed Curley's wife, not out of spite, but on accident. Lennie runs away in fear as a result. He did a "bad thing." The group of ranch hands runs after Lennie in order to repay the act with another murder: lynching Lennie. Bringing the novel full circle, Lennie runs to the Salinas River (where he is found at the beginning). George has run after Lennie, but steals a deadly weapon on the way, and George does this on purpose (and knowing what the mob is about to do). George and Lennie are best friends.
A guy needs somebody―to be near him. A guy goes nuts if he ain't got nobody. Don't make no difference who the guy is, long's he's with you. I tell ya, I tell ya a guy gets too lonely an' he gets sick...
During the entire book these two best friends dream and plan their farm together. George loves Lennie and doesn't want him to die at the hands of an angry mom. This is why George first asks Lennie to talk again about their farm-dream, but then takes the gun and shoots Lenny at point-blank range in the back of the head. In this way, George proves his loyalty, allowing his friend to die in happiness instead of confusion at the act of an angry mob. That, then, is the answer to the one question your submission could mean.
In regard to the second question of why different interpretations happen, the answer is very different indeed. You see, I am a fan of the former interpretation: the interpretation of loyalty and that the dream is gone without Lennie. However, there are other interpretations because Steinbeck purposely leaves these thoughts up to his readers. Yes, George grieves over the loss of Lennie, but some people think that Lennie was truly in the way of the dream and that George could really achieve the dream "better" either alone or with another ranch hand (like Slim). These scholars say that this is the reason why George walks off with Slim at the end: to continue the dream. Another interpretation is that Lennie was getting in the way of George's happiness. This new "error" in killing Curley's wife was WORSE than killing an animal. These scholars portray George as fed up, hence the reason his hand shakes at the end: not out of grief, but out of necessity.
So, you can see that in this educator's opinion, George shot Lennie out of love and friendship and that the different interpretations exist because of how Steinbeck ended his novel. It is an amazing aspect of an author's style to leave an ending that ambiguous (so much so that scholars fight over the reasoning)!
Lennie runs right to the pool of the Salinas River (which is where he was at the beginning of the novel, actually). George meets up with Lennie there (but not before he grabs a pistol). In an act of loyalty, George asks Lennie to talk yet again about their dream of owning a farm together. As Lennie does the recitation of the dream, George's hand shakes as he pulls the pistol's trigger and shoots Lennie in the head. When the rest of the hands arrive, George says that he took the gun FROM Lennie and then shot him. George walks off with some of the other ranch hands. Thus ends the book.