Difficult question! My guess would be that there was not a single mechanism responsible, nor did it happen from one day to the next.
Important mechanisms to name a few probably were:
1)The spontaneous creation of complex organic molecules from simple molecules in proto-earth.
2)The 3D complexity of very long organic biomolecules, that in combination with external energy can create complex chemical systems that are difficult to reach equilibrium. Also these molecules can react with simpler compounds (CaCO3, aminoacids, etc) and create other molecules with ridiculously complex 3D spatial conformations like them and increase the available for life chemical reservoir.
3)The natural history of reactions favors (and in a way select) the most stable systems. For example adhesive properties can make systems endure external events, hydrophobicity can create membranes, pores, etc, which can protect and sequester reactions, deoxyribonucleic acids and their packaging can lead to relatively stable structures, etc.
4)The fact that we are the observers of the whole phenomenon gives us a specific viewpoint that can be deceiving. I will explain:
a)We don’t realize the unfathomable amount of chemical reactions and events that happen even in a tiny bit of living material. Every chemical reaction in life happens for a natural reason. There is locally no purpose in them, they are natural events. However, we dont see plain reactions, we see for example mitochondria, cells, etc.
b)Life from a strictly biochemical perspective is a sum of chemical systems that interact. From a biochemical standpoint, there is not such a thing as an individual organism. This is probably a result of a need of human brain to organize and understand systems. Thus, we think that living systems can create order, while in fact life as a whole, if we even include for example death, food and nutrient recycling is a much more disordered system overall, exactly like all spontaneously forming chemical systems.
c)Life’s chemical reactions seem to have a purpose because life self-organizes. However, who is the observer? We. The results of this process are the observers of the process. Everything that happens leads to them. This by itself cancels out the epicness of self-organization. If a river could think, how would it perceive the cycle of water it participates? Imagine a series of events: A->B->C…….Y->Z->A->B…etc and the observer is (N+O). The observer will think that this system can self-organize. Although an over-simplification because life is consisted by unfathomable numbers of chemical interactions, all i am saying is that any system will be perceived to have self-organizing properties from the perspective of its results.
And similar systems under the same laws of nature will constantly produce similar results. Imagine you go to a planet with many chemical reactions on its surface. Your chemical analysis will be similar both now and after 30 years. Every chemical reaction in life happens for a natural reason, for example adenine and thymine form bonds during replication. There is no purpose in them, they are natural events. DNA molecules are a part of the soup, but they are stable and their interactions will be relatively preserved and will expand. Changes in DNA sequences will affect the fate of the chemical system and the most sustainable results will dominate. Natural selection will be in full action, but its like looking it from a different angle. Biology and chemistry look like 2 sides of the same coin, but it depends on the angle you observe the phenomenon.
d)There is nothing more tricky than to try to understand logic and human cognition by using…human cognition. Its like trying to see your eyes with your own eyes.
All thoughts, regardless of how deep or complex, can be reduced down to simple chemical processes. There are no exceptions.
Reason is driven by what we perceive as pleasure-seeking integral tendencies and internal forces and instincts (reproduction, survival, etc).
If you are starving for days, you are gonna constantly think of ways to feed yourself. Just write down your thoughts 1 day after you quit smoking and read them after 2 years. One can deliberately fast, but that is because the person decided that this will give him/her even more pleasure (e.g due to religious views, etc) than eating.
Logic is also driven by self or other parties aims and interests. They build and use their existing background that is written in nervous synapses and based on what they have learned so far, they seek of solutions or develop strategies to fullfil their aims and interests. They do so, because this will give them pleasure and avoid pain.
In other words, cognition and logic is a tool of nature to serve its purposes. It is a homeostasis tool, and its no different than the lungs, the kidneys, etc on that sense.
So to conclude, there are probably several mechanisms that enabled life to emerge, but the main mechanism that made biology different than chemistry is probably the fact that we are the observers.