The reason for your colleague's departure is probably less important for you compared to how their departure might affect the work that you do. So I suggest to refocus on managing that change instead. At the bottom of this response are some questions to help consider how you might start the conversation with your immediate boss and team members about these issues (except the question about promotion -- keep that one to yourself ;).
My approach to work is asking fewer questions not directly related to the work at hand, while keeping an ear out for possibly valuable information.
In this case, ask yourself, Do I really have a need-to-know about this? Do the specific reasons for this person's departure matter very much to my work?
My guess is no and no. You probably have enough on your plate as a new employee as it is, and could spare yourself the extra anxiety associated with potentially uncomfortable, subjective answers, especially if it's hard to tell if they are the actual reasons, or a 'version for the press'. While perspective is not a bad thing, sometimes keeping the mental blinders in place helps to focus on your own work and avoid the extra drama which is neither necessary nor beneficial.
Regardless, chances are that (if not before then shortly after the person leaves) someone will mention the reason to you or you will have an opportunity to casually ask someone who might know why this person left.
Finally, does it really matter if you knew the reason before the person leaves?
If the answer is no, then you might as well spare yourself and the person the potential awkwardness and wait.
More importantly, staff comings and goings inevitably spell change for those who stay. This change can either go smoothly or make for a bumpy ride during the transition.
What will or might have an impact on your work is not why the person left, but what the fact of their departure means for you in a very practical sense. In this regard, here are some questions to consider now, so that you can prepare and position yourself well for this change:
What does this person's departure mean for the ability of my team to perform its key functions and support mission-critical projects? Are we loosing expertise in a particular area? How easy or difficult might it be to replace this person with someone equally competent (or better)? How long might this take? How might I be able to step in and assist in this transition, and help my team continue to function smoothly once it happens?
How might my workload change before a replacement is found? How long might it take to train that replacement until the new person becomes as effective at the work as the person who is leaving? Until that happens, might I be asked to pick up any of this person's projects, or to help train the new hire? How will this impact my ability to perform the projects that I am already working on?
Will this change team structure/dynamics in such a way that might create risks or open up opportunities for me (or both)? How might this affect my opportunities for promotion?
What might I do now to help minimize any negative impact of this transition? How might I prepare in advance so I am not hit with surprise work assignments later, including things that might not be well documented or I might not well understand?
Approach the situation with a "what can I do to help" attitude, be proactive, and don't be afraid to jump in to fill the gap created by your colleague's departure. There is a good chance that your boss will appreciate this attitude and that you are focusing on the work, as opposed to spreading rumors or freaking out for no good reason. Good luck!