Let's look at this code:

1 if (searchBox == null) {
2     synchronized (SearchBox.class) {
3         if (searchBox == null) {
4             searchBox = new SearchBox();
5         }
6     }
7 }

Let's try to reason about this. Let's say we have two threads A and B and let's assume that at least one of them reaches line 3 and observes searchBox == null is true. Two threads can not both be at line 3 at the same time because of the synchronized block. This is the key to understanding why double-checked locking works. So, it must the case that either A or B made it through synchronized first. Without loss of generality, say that that thread is A. Then, upon seeing searchBox == null is true, it will enter the body of the statement, and set searchBox to a new instance of SearchBox. It will then eventually exit the synchronized block. Now it will be B's turn to enter: remember, B was blocked waiting for A to exit. Now when it enters the block, it will observe searchBox. But A will have left just having set searchBox to a non-null value. Done.

By the way, in Java, the best way to implement a singleton is to use a single-element enum type. From Effective Java:

While this approach has yet to be widely adopted, a single-element enum type is the best way to implement a singleton.