The short answer is that Lennie cannot survive on his own. In chapter one Steinbeck describes Lennie as a "bear", with with "huge paw(s), and as a "snorting horse". These images are meant to convey wild and unharnessed power suggesting that Lennie operates on simple instinct, not reason. This impetuousity is often hazadous.
When George and Lennie arrive at water source that had the potential to be stagnant, George demonstartes caution while Lennie plunges into the water and drinks deeply even though this made him ill only a few nights before. Lennie's instinctive reactions are often dangerous to others as well. One need only consider his history in Weed, with various pets, in mangling Curly's hand, and killing Curly's dead wife.
The dream that held them together died in the barn with Curly's wife. George could not watch over Lennie anymore. Certainly not while on the run from the authorities. Yet, without constant supervision Lennie was a danger to himself and a threat to others. Without George to care for him Lennie would undoubtedly experience and create great suffering before facing his own inevitable death. Candy once confided to George that he regretted allowing another man to put his down his old dog. Similiarly, George felt it was his job to put Lennie down in the kindest possible way for George's good, his own good, and the good of others. George's act is the highest form of mercy.
The answer to this question can be found in the fifth chapter or section of this incredibly moving novel about friendship, loneliness and dreams. Having discovered that Curley's wife is dead, and that Lennie is absent, George is quick to work out what has actually happened. However, unfortunately, even though Candy recommends that they ought to let Lennie just run away because if he is caught by Curley, he would be lynched, George knows that this is not a possibility for Lennie. Note what he says:
"Guess... we gotta tell the... guys. I guess we gotta get 'im an' lock 'im up. We can't let 'im get away. Why, the poor bastard'd starve."
Unfortunately, the reality of the situation is that Lennie would not be able to look after himself if he did live on the run and would probably kill himself through starvation or some other form of accident. Thus George has no choice but to commit his mercy killing.