Garvey is explaining to Cole that he cannot continue to maintain his angry attitude if he wishes to live on the island and be successful. The idea that there is "not room for both" stems from the notion that either Cole can live on the island without his attitude, or his attitude can remain and Cole will be, in essence, removed from the island (physically or in spirit).
Cole's refusal to comply with this notion is demonstrated throughout his first stay on the island as he allows himself to be controlled by his anger. He sets fire to his hut and all of his belongings, which shows his irrationality at that point and his inability to grasp the gravity of his situation of being isolated on the island. As he continues to convey that anger by trying to swim away and to attack the Spirit Bear, we can see his physical self begin to waste away as he allows his "attitude" to prevail. Ultimately, his body is destroyed and there is "no room" for him on the island anymore because his wrath caused him to be mauled by the bear and thus unable to continue living in isolation.
In contrast, during Cole's second stay on the island he has effectively rid himself of the "attitude" that Garvey warned him about. He is able to move past his anger, eventually successfully performing an "anger dance". As he learns about himself and his strength in being able to keep his composure, he uses his new outlook on life to help renew the mental state of Peter, the boy that he originally attacked. Thus, Cole is able to rid himself of the "attitude" and remain successfully on the island.