Like a lot in life it stems from insecurities and a lack of comfort with, and belief in, the self.
I have had this happen a number of times, though for now it has ended up that the individual can't hang.
On the other side of things there are those who get it and simply are impressed when you pass them. This is how I would feel, even though I generally run in the 6:00-6:30 range, so it wouldn't take anything superhuman to pass me.
One of my favorites was passing an old Brit who was a 14:40s or better guy in his day who said, "not many people pass me on a run," even though he was in his 50s and I my 20s at the time. We chatted and I came to learn about his personal bests. I bid him adieu after perhaps a half mile or mile together.
When others say something I remind them you never know how far someone's going on that day or what their overall training is. It's a metaphor for life that you work on you and you can't be overly consumed with the other guy. The keeping up with the Joneses has its negatives.