I think one of the conclusions you can draw from your chart is that older adults are more frequently diagnosed with cancer as a disease. Cancer is an accumulation of genetic abnormalities due to genetic instability, and so for children this occurs during embryogenesis (there are caveats), and we can see that these cancers seem to occur in cells that are quite active during development. We can actually see that adult cancers seem to occur in established tissues at an older age (probably due to inflammation, check out SITC's new book on immunotherapy and cancer).
So you can do this with cancerresearchuk's website: the stat for childhood cancers is that brain tumors and leukemia occur significantly more frequently, which actually differs in overall distribution from the world significantly (ref).
Figure 1. Cancer incidence in children 0-15 by type.
If you stratify by age by those particular cancers, it's much less of a flat line in the 0-15 age bracket. Explaining why is probably a different question all together. The distribution of types of cancer is different for children than for older adults. Thus, the comparison is made difficult because this isn't a straightforward comparison. The best answer is that certain cancers are actually more frequent in children than in older adults and vice versa.
You can start here. Some of the things you will find interesting:
Figure 2. Hodgkin Lymphoma, incidence by age bracket. Interesting to note that non-Hodgkin lymphoma does not occur with any appreciable frequency in children or young adults!
Figure 3. Leukemia, incidence by age bracket.
Figure 4. ALL, a type of leukemia, incidence by age bracket. Based on the data in the OP, you would expect incidence to be highest for 60-74yr olds, but this is not the case.
Figure 5. Brain/CNS tumors, incidence by age bracket.
Figure 6. Bone Sarcoma, incidence by age bracket.