How then do we get from a prebiotic broth to living things?
The honest answer is: We don't know. The scientific method requires that we come up with a well-founded guess (that is, an hypothesis), not in conflict with known physical and chemical rules that explains why all those pieces in the junk yard self-assemble and eventually produce a rudimentary self-replicating robotic rover. Clearly, this is difficult to envisage.
Some think it is so improbable that they deny the possibility of deriving living things from non-living matter. The original skeptic was the famous English physicist William Thomson (1824-1907) (later Lord Kelvin, eponymous with the Absolute temperature scale.) Kelvin (in 1871) pronounced that Life could never arise from non-living matter under any circumstances. (He also declared that the Sun could not possibly have the age required by some geologists for evolution to proceed. This well illustrates the fact that ignorance will not necessarily deter famous physicists from making pithy and catagorical pronouncements.)
Perhaps we have to rely on outer space after all? A great variety of prebiotic organic molecules are seen in molecular clouds in space. (They modify the light traveling to us from stars behind the clouds). It is now suspected that such molecules are delivered to Earth in asteroids, comets, micrometeorites and interplanetary dust particles. Shielded by the minerals surrounding them, the molecules can survive the fiery transit through the atmosphere. Especially one class of meteorites, the carbonaceous chondrites, have turned out to be excellent vehicles for delivery of such molecules.
There is nothing wrong, then, with expecting the delivery of a variety of complex molecules from space - the nuts and bolts of life, if you like - in our junkyard analogy. However, for the reasons mentioned, the delivery of "instant life, just add water" is not a likely option.
Two fundamental requirements are commonly called for when speculating about the origin of Life from life-less broth. One is an organizing principle. In the absence of organic molecules catalyzing other organic molecules (which is how Life makes living matter), how do we catalyze complex carbon chains? The other is a selection process. In the absence of Darwinian natural selection (which operates on varieties within species) how do we get a separation between "useful" and "useless" molecules? Those molecules that represent stages toward the origin of Life, would, of course, be considered "useful".
A number of creative proposals have been made to bring organization and natural selection into the picture, to help make, sort and concentrate useful organic compounds. Interaction with ordered mineral surfaces is an important element in scenarios where clay minerals and other minerals can act as templates and scaffolding for certain types of organic compounds. Sorting can conceivably proceed on the basis of stickiness of molecules. Wills and Bada (2000), for example, suggest that in the nearshore environment collections of molecules that could best resist being swept away by wave and tidal action would have accumulated on rocks and sand grains. The quality of stickiness would have increased through time, by selection, thus favoring the "survival" of more complex molecules.
The main problem is replication. The Russian biochemist Alexander Ivanovich Oparin (1894-1980) who first seriously pursued the matter of how to produce primitive living organisms from carbon and nitrogen compounds emphasized that what is "coacervates", colloidal aggregates held together by electrostatic forces. Enzymes, that is molecules that catalyze synthesis and lysis of proteins, were needed to speed reactions within these proto-life units. Oparin envisaged a struggle for rate of growth between different types of molecular associations, with the prize going to the team with the best combination of enzymes. As a coacervate droplet grew, it would become unstable and break apart to make offspring of roughly the same mix of components.
Replication is not enough, though. It has to be sufficiently true to the original. Life arose when the various collaborating molecules "invented" blueprint molecules whose task it was to make sure that new coacervates splitting off from old ones would grow in a pre-determined manner and resemble their parent. Actually, an alternation between generations is conceivable, whereby the offspring is complementary to the parent and is the template for the next generation, securing identity of offspring and grandparent. Eventually, in the struggle for resources, certain collaborating primitive life forms would win out and make the first organisms, protecting themselves with a secretion, a cell wall. From there on, the powerful mechanism of Darwinian selection takes over in earnest, in the familiar manner.