The "intuition" that he used came in the form of the assumptions used to approach the theory, not in interpreting how it applies to real, physical situations. As a theorist, Einstein had a particular idea about what makes for a "beautiful" theory. His primary idea, as it relates to General Relativity, is as follows:
1. All physical laws must be completely independent of the observer's frame of reference.
2. Gravity is equivalent to acceleration.
Through these basic assumptions, and with a lot of frustrating work and some help from mathematicians, he came up with General Relativity. If he had an "intuitive" understanding of the theory, after all, it wouldn't have taken him more than ten years to develop the theory after he published his Special Theory of Relativity in 1905. It is, after all, no more than an extension of the special theory. It's just a difficult one. He said, in short, "I think the world must behave in this way. Now to do the hard work and figure out precisely what that means."
This is the way that theory progresses. A scientist has a general intuition that making use of particular sorts of rules makes for more beautiful mathematics. Different scientists have different ideas for what makes for the better rules. Some get lucky.
We might argue, of course, that those that end up being correct don't get lucky, but instead are applying some deep physical intuition. I just don't think there's much evidence to support that hypothesis, as at any given time there often seem to be many valid approaches to an unsolved problem. Though having that intuition may help narrow the playing field, it won't narrow you in on the precise theory.