It might be useful to the intuition to consider the following simpler strategy: If the revealed number is positive, guess that it is the larger of the two. If the revealed number is negative, guess that it is the smaller of the two.
This simple strategy already guarantees you a chance of winning that is always at least 50% and sometimes greater. In particular, your chance of winning is 50% if the numbers are either both positive or both negative, and 100% if they have opposite signs.
That's not quite a solution to the original problem, but it works --- or comes close to working --- for exactly the same reason that the actual solution works, and it's easy to understand in an instant.
