To properly understand how traceroute works one would need to know about the TTL field

I did learn about this on my own that day, but the original confusion was at a quite different level: I asked whether the times on each line measured the distance between that router and the previous one, or between that router and the source. His answer: "Both." A charitable interpretation of this would be "They measure round trip times between the source and that router, but it's just a matter of arithmetic to use those to estimate round trip times between any two routers in the list" -- but I asked him if this was what he meant and he said no. We went back and forth for a while until he told me to just research it myself.

Edit: I think I remember him saying something like "You're expecting it to be logical, but things aren't always logical".