It shows how the rate of growth (cm/day) on the vertical axis varies with the ice thickness (horizontal axis). Say you increased the length of the freeze season, you'd get only small increase in thermodynamic thickening because by 1.5m and above the rate of ice growth is small. This is because as the ice thickens it better insulates ocean (warmer) from the colder atmosphere.
This is quite interesting, but wouldn't the same/similar effects likely occur in reverse in the summer, i.e. that the ice tends to melt faster the less of it is there is? It seems like your argument sort of boils down to the idea that it's easier to form new ice than to melt it, but is that what the underlying physics really suggests?
No, no. You're getting it all wrong. You see, graph presented is about a case when there is no insolation to talk about (at the pole, this starts as early as ~20th September). In such conditions, after several dozens hours, the air naturally cools down to much below 0C (as it radiates some of its energy into space, and there is no Sun to warm it up in a daily 24h cycle). In such conditions, there is a big difference between air temperature (which is much below freezing point) and directly-at-the-surface water (which, because of its huge thermal capacity and convectional mixing cools down dozens, if not hundreds, times slower - and sometimes remain liquid even at surface for quite a while after the Sun goes below the horizon for the polar night). The surface of the ocean is rarely exactly plain as long as it's liquid water - because of winds, there are nearly always some waves, quite often big ones. Mechanical motion creates mixing, and for a while, this prevents ice layer from forming. Not for long though, and depends on conditions. Low/no wind and/or snowing eventually happen, and this is the start of ice formation. Once even very thin layer of ice has formed, THEN you start to see what the graph is about: very rapid growth of ice initially, provided very cold athmosphere (below -20C) remains (which is usually the case).
But the thicker ice gets, the better insulator it becomes: i.e., the less thermodynamic interaction there is between cold athmosphere above and liquid water below. That's why the growth slows down the thicker ice gets. The graph importance is _how_ - quantatively, - this process happens "with everything else being the same". I.e., if the water udner the ice suddenly changes to be much warmer than near 0C - then the graph is irrelevant; won't describe things happening. If the air above gets above 0C, or when polar night is not the case anymore - i.e., it's spring time, - the graph is also irrelevant.
And no, the graph presented does not apply to melt. During melt times, there is the Sun, which is very, very major factor - deciding factor; there is secondary heat from the Sun - re-radiated by air, by GHGs in the air, by water in any currents which go directly under the ice and even - to lesser extent, - from deeper yet warm currents (Gulfstream remains in the Arctic are known to get somewhat deep at places, for example), etc. Melt is happening completely differently. It's more like "pulsing mode", instead of "rapid first, slows to crawl later" or "crawl first, accelerates to rapid melt later". There is, if you will, "breathing" when Arctic melts - it goes in somewhat rhytmic big chunks, quite often. The main "initiating melt" mechanic is, afaik, warm water and air currents (significantly above 0C), which "intrude" into cold enough "so far" areas. Air currents warm them up to the point of melt ponds on the surface (albedo plummets - direct insolation starts to work full-time). Water currents warm them up "from below", decreasing thickness of ice and fragmenting ice fields into "chunks", with bits and pieces of open water (which does not refreeze as long as warm water current is there), and then the more open water there is, the more direct sunlight absorbtion kicks in. When strong warm air and water currents "attack" an ice-covered region, it often "bursts" into open water incredibly fast. If it's clear sky and the time is May...August, then it can be amazingly fast.
Where those warm water and air currents come from? Warm water comes from already ice-free regions, and for shore - from the rivers, too. Warm air comes from same places and - later in the season, - gets very warm in the Arctic itself whenever it's not heavy clouds in the region (especially with higher and higher GHG content, it happens faster and to higher temps).
A small fraction of sunlight gets absorbed by the snow and ice too, but i believe it is a secondary melt mechanic.
So you see, the freeze is driven, primarily, by local process: the air "here" gets cold (without the sun), and it freezes sea surface "here" to ice; but the melt is primarily driven by inter-regional processes: the ice "here" is cold and it's reflecting most of sunlight, not allowing water below to warm up and only slowly warming up itself, - but water and winds from "somewhere else" come in, and do the lion share of melt. Of course melt can't be defined by a graph which defines freeze - melt is much, much more complex thing, since it's not a local process.
I hope i helped.