Justifying past dropping out in statement of purpose letter
The overriding goal in this situation is to communicate in a way that causes the admissions committee to believe that you have the qualities to get through a PhD program, and that you will be someone who is easy and pleasurable to work with.  It is therefore important to communicate in a way that gives cues as to your fortitude and drive to get through the program, and shows that you are calm and able to deal with conflict in a mature manner.  Anything you say that detracts from this could be detrimental to your application.


  Do I need to mention/elaborate on the reasons that made me drop out? If so, what is the best way to express them without hurting my chance of getting an admission?


This is certainly a thorny situation.  You will certainly need to give some explanation of why you did not succeed in your previous attempt at a PhD program.  However, unless there is some simple, clear, and objective evidence of misconduct by your previous supervisor (e.g., a disciplinary outcome against him by the university for his treatment of you), it is not a good idea to go into specifics, or make statements claiming psychological abuse.  The problem with doing this is that the committee will have no simple way of confirming that your claim of abuse is accurate, and it will therefore naturally give rise to the alternative possibility that you are just extremely sensitive, and perceive psychological abuse in ordinary supervisory activity.  I am not suggesting this is the case, but just bear in mind that the committee will not have any way of confirming your assertions, so they might consider this to raise a risk for them.

In this kind of situation, assuming there is no simple, clear, and objective evidence of misconduct by your previous supervisor that you can cite, I recommend that you include a short statement to the effect that your relationship with your previous supervisor was "not a good fit", and this hindered your progress.  The committee is likely to "read between the lines" and understand that there was some problem between you and your previous supervisor, so by understating the problem (e.g., saying it was "not a good fit" without blaming anyone), you signal that you are a person who can roll with the punches.

If you get an interview for a prospective PhD candidature, be prepared to field follow-up questions about this issue.  I would recommend preparing some verbal answers that take the same understated tone, but also be prepared to give full disclosure in a calm and measured way if pressed on the issue.  If this comes up, you again want to communicate that there was a good reason for your previous failure to complete, but you have the fortitude to talk about it in a calm and non-accusatory manner.