Many animals and humans equipped with appendages for locomotion exhibit a stride in their movement, as when a person is walking, for example, and he takes extra-long steps, as if he is impatient, in a hurry, or upset about something. If his stride is broken, it might be because something has distracted him, or someone grabbed hold of him and "broke his stride."
Similarly, a horse when running at a race track has a certain stride, and it can be measured in feet and inches (meters and centimeters) by measuring the distance between when all four of the horse's hooves leave the track and when the hooves contact the track once again.
If the horse were to be frightened (or "spooked") by something, he might break his stride, and the distance I've described above would shorten considerably. In fact the horse might simply stop galloping and stop altogether.
Breaking Stride versus Catching Off Stride
With these two illustrations in mind, we can see how natural it is to apply the phrase "break (or broke, breaks, breaking, broken) metaphorically. Breaking stride happens at the point of time when something or someone breaks it. Being caught off stride occurs when the person can trace his or her being sub-par to an event in the past. When your concentration has already been broken, for whatever reason, and a person asks you a question, you might say to the questioner,
"I'm sorry, you've caught me off my stride."
In other words, something has occurred to a person in the recent past which causes him or her to be at less than optimal, mentally, physically, or emotionally.
Breaking Stride
So, breaking stride occurs when there is an interruption. As soon as the person is interrupted, she or he is off his or her stride. A questioner in this scenario may not likely get a well thought out answer and would do well to hold the question for a later time. On the other hand, some questioners like to see people squirm once their stride is broken. In other words, they move in for the kill! Not a pretty sight!
A couple illustrations might help. A Secretary of State who is involved in tense negotiations with the representatives of two countries which are at war with one another might be making good progress in forging a peace plan, when all of a sudden, a bomb goes off in the building where the negotiations are taking place. The bomb broke the Secretary's stride, and the negotiations ended then and there.
Similarly, whenever someone is making good progress, she does not want to break her stride, because she sees the light at the end of the tunnel; the goal is in sight.
On the other hand, when a reporter at a presidential press conference hears the words "white Republican," all of a sudden the rhythm of the conference for perhaps that one reporter only is interrupted, because the words startle him and he almost loses his train of thought. Consequently, his mental stride is broken and he needs to deal with the interruption--process it--before he can regain his stride, metaphorically speaking.
Having one's stride broken can be a frustrating occurrence. Again, think of the person who because of urgency, haste, impatience, or anger (or a combination thereof) is making good progress towards a goal, but all of a sudden her concentration and focus are broken by some irritating interruption. The very real temptation of a person so interrupted is to quit, to throw in the towel, and try to regain her stride sometime later.
Caught Off Stride
Being caught off stride may have little or nothing to do with the present situation; rather, it could be traced to something which happened earlier. Consequently, when a politician, for example, is unable to give a reporter an answer to a question, the politician might say, in effect:
"Mr. Smith, I appreciate your question, but because I just emerged from very tense negotiations with the ambassador to North Korea, my mind is elsewhere and you've caught me off stride."
In conclusion, a weight-loss dieter might be making tremendous strides in taking off the kilograms (pounds), but one day he gives in to temptation and pigs out at an all-you-can eat buffet. Now he might take it in stride and simply continue the weight loss regimen the next day, OR he could give up the diet altogether and start overeating and under-exercising. Pretty soon, his weight is back up where it was (or even higher!).
There is obviously a psychological component to achieving a stride and then having it broken, as well as to the decision to give up or to persevere, once you regain your stride. Leon Festinger's concept of "cognitive dissonance" comes into play here, but I'll reserve that for another time, because it might break your stride in understanding breaking stride!
Best Answer
Essentially, it means not being able to resist forbidden temptations. You are told not to take any more cookies from the jar, but you can't resist. That's a metaphor for all the other temptations you are supposed to avoid.
(By the way, you can google these kinds of phrases and find lots of ready-made answers.)