Because those are the bounds of the data. There was no attempt to hide this fact in the graph (plot does not intersect "origin", well labeled axes...)

It's really not. "[Mauna Loa Observatory] was founded on June 28, 1956, as part of the US Weather Bureau"

And if you look at the tick marks on the graph, it starts before 1960.

It starts at 1958 because that's when the measurements began.

MLO was founded in 1956 and began continuous CO2 measurement in 1958.

>Why does the Y axis start at 320?

Because CO2 didn't start from zero in 1960?