Imagine I'm using pi = 3 (accuracy of 1 significant figure) that's an error of about 4.5% of π(pi), 3.1 is only 1.3% 3.14 only 0.05% with the error decreasing with each additional significant figure.

Imagine there's a circle with radius 1m and you've got a calculated bearing to it calculated using pi = 3 . In the worst case in 2 dimensions for every meter you walked you could be walking off to the side ~0.0225 meters (napkin maths) from where the circle really is, so it would only take the circle being ~45m away for you to walk right by it rather than through it. With pi =3.1 you're diverging ~0.0066m per meter so the circle would need to be ~152m away before there was a chance you'd miss it. 11 digits of pi gets you about 3 light years of walking before you had a chance of missing.

They were discussing (with a large degree of understatement,as discussed by others above) that this value of pi gives great precision in these kinds of calculations.