One of the suggested possibilities is that there was a pre-RNA molecule, such as PNA[1]. PNA is less efficient but it's much more durable, simpler to form, and can self-polymerize above boiling. If you had enough PNA in one place and got it relatively hot, it would spontaneously form "genes" which would be stable for an extremely long time. They would just float around, and any time they came into contact with enough PNA they would be able to replicate. It would be a relatively slow process, but purely random and so would quickly gravitate towards optimization.

RNA can do the same thing. Transcriptase and polymerases are catalysts but they are not strictly necessary. Simple RNA can have a lifespan of hundreds of years and it just needs to replicate more than once to propagate. Over time a primitive polymerase would eventually be made by freak chance and mutation, and then everything would kick off.

Once you have polymerase natural selection would speed up massively and become resource-limited rather than reaction-limited. Thats when a metabolism would evolve, and enzymes that were successful at catalyzing the conversion of material into resources would stimulate local growth. You'd still only have a frothy mess of chemicals, but it would be a frothier mess than the surrounding mess. The first organism to make some kind of container wins- most likely they'd evolve mucus-y proteins that would help keep their resources concentrated. Over time that mucus would give way to a hollow bubble, which would then become a lipid layer, which would then become the first cell.

[1] https://en.wikipedia.org/wiki/Peptide_nucleic_acid