Yes, it's a lot like existing solid-state nucleic acid or peptide synthesis setups, but with the major advantage of forming carbon-carbon bonds instead of phosphodiester or amide linkages, making the technique a lot more general. The setup involves a useful reaction called Suzuki coupling. In Suzuki coupling, a metal (usually palladium) catalyzes a reaction between a halide (that is to say, chlorine, bromine, etc.) and an organoboron compound. The mechanism is complex, but the result is a carbon-carbon single bond. This reaction and similar ones are already widely used in the pharmaceutical industry since they can reliably glue together smaller structures together to make a larger molecule. The smaller structures are not individual atoms, though- they tend to have maybe 10-20 atoms or so. Drugs with biaryl structures like the blood pressure drug valsartan are now often made this way.
In previous work, the Burke lab showed that the reaction could be made more convenient by using a specific type of boronate salt which can be easily added and removed from a molecule, and generally produces derivatives that are stable long-term. They then found that these salts can bind to silica and will only be released in the presence of the solvent tetrahydrofuran. So what they did was build a setup that can run this reaction iteratively; at each step, you add another bit of the molecule; each bit has a halide at one end and a boronate salt at the other. This is a lot like an amino acid, which has an amine at one end and a carboxylic acid at the other, which can each react with other amino acids to form chains. Since the molecule bits are shelf-stable, conceivably you could load a machine with a library of commonly used "puzzle pieces" (which you probably bought from a specialty chemicals manufacturer like Sigma-Aldrich or EMD) and assemble them, then wash off the finished product in THF. The yields demonstrated thus far are...not great, but the idea that it can run automated means that it could brute-force some syntheses and allow for the production of complex molecules from more common starting materials. It's a major advance in synthetic organic chemistry, but it's not so much a universal printer as more like an early mechanical printing press, where you still need to provide the type blocks and set the letters yourself.