I just relax and think logically. I assure myself that I am in control and no matter what happens, it cannot be that bad. The only situation where this doesn't work for me is with medical and dental stuff. I have an extreme needlephobia. As for why some people are better at dealing with it than others, it depends on a lot of things. Personality, upbringing, environment, social situation, and lifestyle all play in. I am sure that the APA (American Psychological Association I think) has many articles on this subject in varying degrees of detail.

Zen practice has completely altered the way I relate to stressful situations. Before, I would try to control the situation using strategies developed over a lifetime. That might mean (for example) avoiding conflicts, or over-preparing, or allowing myself to slip into "white lies" to defuse problems. For a long time I dealt with the strain of daily life by drinking too much, also. Now I mostly just allow whatever is happening to happen. Things come and go: situations change, feelings and thoughts arise and depart, life flows on no matter what we do. That doesn't mean I don't take any action, on the contrary I find that my actions are much more appropriate and responsive than they used to be. It just means that I'm much less likely to take things personally than before. Most of what we call "stress" is actually just habitual, repetitive, self-centered thinking. Our egos are engaged in an endless and futile battle to maintain their stability and boundaries in the sea of unending change and flux that life is. For those fortunate enough to see that the ego is really just a psychological construction with no substantive basis, this preoccupation with self-construction and maintenance drops off to a reasonable level. That leaves the mind free to experience life in a much richer way.