In a genetic algorithm, crossover (recombination) is the analogy to mating in the real world. For example, you have some genetic information from each parent. In the case of an optimization where you have vectors of features (design variables), you could represent it as vector 1 and 2. Imagine each vector has 10 values. You grab the first 5 from vector 1, and the second 5 from vector 2, and combine those into a new vector which is your "offspring". Note that there are many different ways to do recombination.
When you are deciding on parents to use for recombination, you want to keep some of them the same. Recombination is basically hoping that combining attributes will give you a more optimal set. However, sometimes, you might want to make some of the offspring be copies of the parents in the next generation. So, as this link explains, the crossover probability determines what percentage of offspring experience recombination. It is probabalistic because there is an element of chance involved in whether or not given offspring experience recombination.