tl;dr- Sounds like something went wrong in your past. Prospective employers would likely be interested in understanding this prior event to inform their consideration about whether or not you'd be suitable for a new role.
Personally, I'd guess that the best way through this is to help prospective employers to understand what happened and how this information should inform their evaluation of whether or not you're the appropriate candidate for the job.
Most of the following's written in the first-person. This is meant to reinforce the point that I can't really speak for hiring managers in general as there're many diverse viewpoints and philosophies. So, the following should be understood as my own thoughts and perceptions. Also disclaiming that none of this is legal advice; for example, there may be relevant laws/regulations/etc. involved in interviewing interactions and disclosures that aren't considered below.
I know the reasons in my head for dropping out of the PhD are sound, but I am struggling to come up with a way that explains what happened without potentially giving off an impression that I made a bad decision.
Sounds like you did make a bad decision at some point. Still, unless you're trying to convince someone that you're infallible, that's not the worst thing in the world. Then on the plus side, you avoided making the sunk cost fallacy: the decision to stick with a path that's no longer optimal merely because resources have already been spent on it.
Personally, if I were considering hiring you, I'd want to know the following:
What mistake(s) did you make?
- I'd be extremely interested in this question, so you'd probably want to help me figure out the answer. If I couldn't figure it out, I'd likely perceive there to be an unqualified risk, and probably opt to avoid that risk by not hiring you.
What do those mistakes say about you at the time you made them?
What do those mistakes say about you now?
- What, if anything, have you done to correct for their causation?
How concerned should I be about any potential problem areas that they may reflect on?
How should I evaluate your response to these various problems?
- While causing problems may reflect negatively on you, fixing them can still reflect positively. You can earn points here.
It sounds like you've already given these some thought. However, you might want to give it a bit more thought. For example:
I hadn't fully developed the soft skills needed to be able to tackle one (e.g. time management, organisation, resilience) which resulted in slow progress.
That sounds pretty bad. And while I'd give you kudos for recognizing it, I'd still want to feel confident that this problem is resolved, or at least manageable.
A major point here is that, if I hire you, I want to feel confident that I made the right decision. So I need to understand exactly what happened and why, despite those facts, you're still the right person for the job.
So, here're a few of my first thoughts on seeing your points. I'm speaking in the first-person here because my perceptions are my own; I can't generally speak for any hiring manager who might be considering you, though I hope that these may be helpful.
- I hadn't fully developed the soft skills needed to be able to tackle one (e.g. time management, organisation, resilience) which resulted in slow progress.
Apparently you had trouble managing your time and working at the expected pace. That's a pretty big problem in most positions in which you'd need to manage your own work flow.
Have you done anything to fix these problems?
Are they still problems?
If I hired you, how should I expect them to complicate things in the future?
- I was struggling to manage myself in a lot of ways (e.g. money, food, mental health)
If you were a PhD student struggling with money or food, that's not so unusual or hard to understand. If you're applying to be a PhD student again, it'd be a big point to address because it seems like you might have the exact same problems this time around, too. But if you're applying for a higher-paid position, it'd seem less relevant (though you might still comment on if it'll be a problem, and if it won't, why not).
As for issues with "mental health", that's alarming. Poor mental health can truly wreck people, and there's often very little an employer could reasonably do to help fix it. No employer wants to be in that position, so you may want to:
Think about all the ways in which your mental health might adversely affect your employer.
Try to figure out how you might correct for any of those problems before going into any interviews.
Explain these to the interviewer.
Note: There may be various laws/regulations/etc. surrounding mental health disclosure, discussion, hiring consideration, etc.. I'm not a lawyer and I'm not offering legal advice. I am not asserting that anything here or in this answer is compliant with any given legality's rules or regulations.
- The area ended up being quite a bit out of my comfort zone (i.e. more pure mathematics compared to an applied mathematical background) and was chosen more based on fanciful ideologies rather than what I was traditionally good at, and it took me a long time to do things that my supervisor thought were straightforward.
Is this a concern for whatever you'll be doing in the new position that you're being considered for?
Does this reflect on any sort of inflexibility that might result in an unwillingness to perform some future job function?
- I didn't have a break before starting my PhD and felt consistently burned out/depressed having gone straight into it after my Master's degree - not the best decision to make in hindsight.
Could this be a problem now?
Are you prone to burning out, such that it might be a problem a few years down-the-line?