You asked for it, Quod.
To not get crazy about it, or to go far back into abiogenesis (at least on the atmospheric level which is the really important factor) which is not something many people are in a position to speak about confidently, it allll comes down to the chemistry and organic molecules.
Probably prompted to some degree by something that was going on in the environment, over time we got organic chemistry. Everything in this realm is an issue of principles of chemistry, the ones that are still observable today. Things that liked to connect connected. Things that were energetically unfavorable typically didn't come together. Organic chemistry is where you start seeing complex molecules, like the building blocks for DNA and RNA, and things like lipids that form bilayers and vesicles.
The phospholipid is something I always like to think of. It's still just an issue of chemistry. But it has one end that is hydrophobic and one end that is hydrophilic. If you put a whole bunch of phospholipids together in water, they will pretty much always fold into predictable shapes on account of polarity. They like their layers. They are happy to collectively fold into a sphere. The phospholipid bilayer that pretty much wraps up your average cell is something so critical to the cell, obviously, but something that occurs so easily.
Similarly, getting to the real crux of the cell (proteins and eventually organelles and maybe someday even some DNA), the most important product of this chemistry will be things like nucleotides, and there are a few examples of molecules in organic chemistry that have ostensibly unnecessary nucleotides stuck to the side if it. In modern cells, we think of DNA as being more or less the language of life, but in the drafting days of the organelles and the cells, it is far more likely that RNA, which is something increddibly simple (relatively speaking), was doing the work, given that to even make DNA is an RNA process all on its own.
RNA sounds like a kind of crazy, complex thing, but it is really still just an issue of chemistry; certain molecules like other ones, and the only thing that really comes out of left field in the early days is going to be environmental conditions. You can kind of still see both of those today. If I had a tube full DNA in an aqueous solution and boiled it, the bonds linking the complementary pairs would be broken and the DNA would denature into two separate pieces. Once I took the tube away and let it cool down, in a time that depends on the ratio of DNA to solution, the DNA would have found its way back together again.
The first cell, undoubtedly, was one that really lacked a lot of sophistication, as there is good reason to believe that at least one of our organelles (mitochondia) were once prokaryotes just doing their own thing.
Tl;dr:
Chemistry, man. Chemistry.
Topic: First Living Cell - How life began