I understand, and have had similar experiences.
I agree with Curtis that it seems to be a form of empathy. Beyond that, I think these things cannot be explained. Of one thing I am sure, the emotion you felt is something genuine, even if I can't explain it.
I can share my experience of when I visited Temple Wood in Arygle, Scotland.
This is a site dating back about 5000 years, with stone circles, grave cists and nearby burial cairns. It's been a sacred site for a long time.
My dad had traveled there years before. It was December, long after dark, and he was crouching against one of the standing stones trying to keep himself and his camera warm while awaiting the moonrise. His companions (Helen, my dad's Scottish girlfriend, and my brother) had returned to the car to get warm. The ground was frozen, so any footsteps would be audible, yet he suddenly realized that there was a child standing a few feet away, staring at him. He said nothing, and just as quickly, she was gone.
When he returned to FL, he continued to correspond with Helen, mostly by sending cassettes. On a tape he received the next summer, she said she was going to Temple Wood the next weekend.
"I wonder if I'll see the wee lass."
He'd never told Helen or my brother about seeing this child. When he replied, he asked Helen to describe "the wee lass." She described exactly what he remembered.
Twenty one years later, I parked in the car park, got out of the car after picking up my camera, and walked into the older of the two circles. Immediately, I had a powerfully intense feeling of...something. I can't characterize it, the closest I can come is some sort of anticipation, or recognition of the onset of something, but I have no reference to compare it to.
When my wife saw the look on my face, she took the camera strap from my neck to take my picture, which grounded me in the moment again. Whatever I was feeling had disappeared. After she took the camera, I tried to relax and let the feeling return, and it did momentarily, but then my dad started talking, and it was over.
Just like you, I've been left wondering, "What happened?" I don't think we can explain it.
When we returned to Edinburgh, I talked to Helen about my experience. Helen was an educated woman, an RN, a midwife, and after retiring, returned to University to study history.
Her theory is that our DNA has a form of memory, and that an ancestor of mine had a connection to that site. I have to give her opinion weight, since she is the only person I have ever known for whom time wasn't a linear event. I offer as one of many examples the time my dad took her to the harness racing track. She picked the first two horses for nine races in a row, but wouldn't gamble because, "It would be cheating."
Or, had I set myself up to react to this place after hearing my dad's story? Does that explain it? Is any explanation more plausible than another? Is it relevant that in the intervening years, the grave of a child had been found at Temple Wood?
All I know is, I need to go back.