Hi Laura,
Your interpretation of the ICC is correct, that it's testing how similar the measures are.
So in slide 42, where you have ICC>=0.93, >=0.99, >=0.97 -- these are all high measurements, saying that the alpha diversity, beta diversity, and the most abundant phyla -- all are very similar between the two extractions. This is probably what we would hope to see given they are duplicates.
Slide 45 takes this much further, measuring over 6 mos for 2 yrs -- to see if they are still high, and they are relatively high, just not as high as before (probably about what you'd expect given much more time between the measurements) with ICC>0.6. It's kind of the same question as are the duplicates similar, but on a much longer timeframe. Although not as high before, it is still high, indicating that the measurements still say roughly the same even with repeated sampling over 2 years. So it's essentially testing if the diversity is stable within the same person over many months.
Finally, in slide 51, it looks to be a summary of Shannon's index. For another reference on Shannon's index:
https://en.wikipedia.org/wiki/Diversity_index
but estimates toward 1 means the abundance of all species are the same, and towards 0 means that it is mostly one species.
Months 7-9, presumably July, August, and September, i.e., the summer months, all show higher estimates, hence why you thought that there was a difference in diversity over the summer months.
Best,
Tom