Why do canned beans have less FODMAPs than dried beans?
Why would canning reduce this process further [than soaking]?

Because canned beans are cooked.  Cooking adds to further breakdown of many flatulence-causing elements.  In fact, if the primary concern is flatulence and indigestion, several authoritative sources have recommended skipping soaking and just using longer cooking.  I'll quote from my answer to a previous question here:

The main reason often cited for soaking is to prevent flatulence. However, if you throw out the soaking water, you also throw out lots of nutrients. Recent research suggests that long slow cooking is a better solution and probably gets rid of more of the flatulence-causing components than a soak followed by a quick cook. And you get to retain more nutrients. To quote Harold McGee from On Food and Cooking:


  One kind of troublesome carbohydrate is the oligosaccharides [which
  are water soluble].... But the latest research suggests that the
  oligosaccharides are not the primary source of gas. The cell-wall
  cements generate just as much carbon dioxide and hydrogen as the
  oligosaccharides--and beans generally contain about twice as much of
  these carbohydrates as they do oligosaccharides.


Based on this research, McGee suggests:


  [Soaking] does leach out most of the water-soluble
  oligosaccharides--but it also leaches out significant quantities of
  water-soluble vitamins, minerals, simple sugars, and seed-coat
  pigments: that is, nutrients, flavor, color, and antioxidants. That's
  a high price to pay. An alternative is simple prolonged cooking, which
  helps by eventually breaking down much of the oligosaccharides and
  cell-wall cements into digestible simple sugars.


Specifically regarding galacto-oligosaccharides, there are studies that have attempted to quantify the role of soaking and cooking. That linked study concluded:


  During soaking, total α-GOS content decreased between 10% (lentil and
  faba bean) and 40% (chickpea). [...] Cooking further decreased α-GOS and
  increased total dietary fibre content.


The explanation for the cooking effect is given later in the article:


  Cooking after soaking led to a further decrease in raffinose (−32%),
  stachyose (−25%) and verbascose (−35%) and to a significant increase
  in galactose content (+54%) in the whole dish. [...]
  This should be attributed to further enzymatic degradation, due to
  better conditions for the expression of α-galactosidase activity.
  Alpha-galactosidase from lentils are active in the temperature range
  20–50 °C and up to 65 °C, and have optimal pH of 4.7, 5.5 or 6.1,
  depending on their isoforms. [...]
  During heating, the temperature increased
  progressively and conditions were met for higher α-galactosidase
  action.


Basically, elevated cooking temperatures increase the activity of the enzyme alpha-galactosidase, which is naturally found in beans and lentils (and is perhaps better known as the active ingredient in many anti-flatulence supplements/medications).

Hence, canned beans that are thoroughly cooked likely get both the breakdown of the cell-wall cements that McGee mentions as well as increased breakdown of GOS through elevated enzyme action.

I would note, however, the detail in the abstract to that study that each type of bean or lentil will likely respond somewhat differently, as each has its own particular makeup of problematic elements (and those elements will be affected in different ways by cooking).  For example, this study on red kidney beans showed significant reduction in lectins, raffinose, and stachyose during soaking, and significant further reduction in lectins and raffinose during cooking.  But cooking provided no significant reduction to stachyose.

Could soaking the dried legumes for a longer period of time and replacing the water a couple times reduce the FODMAPs to the same level as canned beans (maybe in the fridge to prevent bacterial growth)?

Well, as seen in the charts for example in the last linked study, increased soaking time shows diminishing returns.  Replacing the water could help a bit, but refrigeration may also slow down or stop some enzyme activity that could help in breakdown.  (The linked article notes that 77F was chosen for a soaking temperature because it is used by commercial canned bean producers.)

Maybe I should soak the beans, cook them, and then soak them further? 

Again, while this is possible and likely to have some effect, there will be diminishing returns.  And with each replacement of soaking water, you will like lose more beneficial nutrients from the legumes anyway (along with the things you are trying to get rid of).

Or is there something else special about the canning process compared to using dried beans?

Well, the only thing that might be special about the canning process is the high temperatures canned beans are subjected to for sterilization.  In a cursory search, I didn't immediately come up research that quantifies what that might do, but my guess is that it could lead to further breakdown of some flatulence-causing elements.  However, I'm not sure about the effect of high-temperature cooking on galacto-oligosaccharides in particular, as the first article linked above postulates that much of the breakdown is caused by enzyme activity that wouldn't happen at canning temperatures.  So, in that case, it's just the long cooking process for canned beans in general that may have some benefits.