Well if you draw two "straight" lines on a hyperbolic plane which don't meet, they don't stay a uniform distance apart like "sensible" straight lines ought to.

I've put "straight" in quotes because they are straight lines as far as the frilly surface is concerned but you'd look at them and say "these lines are obviously curves". I'm trying to crotchet up a fairly large plane for you and then I'll put some "straight" lines on it and try to explain a bit better - like why I feel justified in asserting that the frill thinks an obviously curved line is straight.