One use of the term Cross Section in the physics of particles moving close to each other is a representation of the probability of interaction between the particles as an area around one of the particles. Among the areas of physics where the concept is used is nuclear physics (fission and Fusion, such as the power generation within stars), in Radiative Transfer, and in the physics of gases as interacting particles (Kinetic Theory Of Gases). The term Collision Cross Section is sometimes used when the interactions are thought of as collisions between particles.
If two particles can interact, one particle passing sufficiently close to the other has a chance of interacting with it. When the point of interest is the frequencies of the interactions within a considerable amount of some substance, this "chance of interaction" between two particles is the same as it would be if the particle had some specific size such that passing close enough to touch results in an interaction. If such a size can be determined through experiment or analysis, it provides an intuitive way of thinking of the interactions that can be useful in thinking about what happens and can be used in some useful calculation as well. Given the probabilities inherent in Quantum Mechanics, such a cross section might be merely theoretical (e.g., considered Quantum-mechanically, there could well be a non-zero probability that one particle passes right through another) but one can still be devised that represents the chances of an interaction.
Given such a cross section, the path of a moving particle can be viewed as a straight round tube-shaped volume, whose calculated volume along with the density of the material allows calculation of properties of interest such as how long the particle is likely to move before an interaction (the Mean Free Path).