Probabilistic reasoning is the opposite of black and white thinking.
When you reason in black and white you ask questions like: Is this true? Is this the right thing to do? Am I sick?
When you reason probabilistically you ask questions like: How likely is this? What's the expected value of this action? What evidence have I seen that I am sick, and what evidence that I'm not? How likely is it that I'm sick without taking any evidence into account?
Reasoning probabilistically allows you to change your mind incrementally, accumulating many small pieces of evidence rather than requiring one overwhelmingly convincing piece.
On LessWrong, 'probabilistic reasoning' usually refers to Bayes theorem, which formally defines the optimal way to change your beliefs when you see evidence.
See Also: Bayes Theorem, Belief Update, Expected Value, Probability and Statistics