It's actually 'Slowly, slowly catchee monkey.' It comes from the days of British colonialism when many far eastern countries were under British rule. Soldiers posted there used to try to catch monkeys to keep as pets and despite all their efforts were not very successful. Many natives spoke 'Pidgin English' which was a simplified form of English. They showed the soldiers how to do it by demonstrating a simple but effective method. This was to place a basket containing fruit in a clearing in the jungle where the monkeys lived. The basket would have a narrow opening and be tied to the ground. Monkeys would come along and stick their arms inside the basket to get at the fruit. The opening was just wide enough to allow for this but as the fruit was grasped the fist was too big to get out of the basket. The monkeys were so greedy that even when soldiers approached they would not release the fruit and thus be easily caught.Naturally this took patience on the part of the captor who would have to wait quietly until the monkey was snared. Thus the slow approach proved to be more effective hence the phrase.
This is now used to describe that a slow and patient approach to a problem with careful thought is often better than rushing in.