For both an acknowledgment of the counterintuitive meaning and an example of the confusion it causes, just head to Wikipedia's article for Transparency (human-computer interaction):
Any change in a computing system, such as new feature or new component, is transparent if the system after change adheres to previous external interface as much as possible while changing its internal behaviour. The purpose is to shield from change all systems (or human users) on the other end of the interface. Confusingly, the term refers to overall invisibility of the component, it does not refer to visibility of component's internals (as in white box or open system).
Other editors object to this as doublespeak and an improper use of the term, so they've flagged the article as factually disputed. You can see a lot of discussion arguing back and forth, as some users apparently reject the meaning that transparent has clearly taken on. But the user at 220.127.116.11 describes both meanings:
At times, transparent is used in the sense that glass is transparent. The details behind the "transparent" glass are clearly visible.
Yet, transparent can also mean the details are obfuscated to avoid confusion. We use this sense when we say, "transparent to the user." In other words, the user enjoys the benefits of a particular function without being aware of how it is accomplished.
In a way, you can think of the details themselves as transparent, i.e. the user "sees through" them as through glass.
I bolded the key explanations above, because I think they accurately describe the way transparent is being used. In one sense, the system as a whole is transparent (you can see how its internals work). In the other, the component is transparent—in this case, that component is the process of changing senior management, and because it is transparent (invisible, really) it doesn't affect the way the customer sees the system as a whole (the company).
Somehow, most people understand which sense is meant, despite the two apparently opposite meanings. Confusing? Probably. But that's English for you.