Think about the transition to the industrial revolution, because that's primarily what Marx was talking about. Prior to the IR, most manufacturing was small scale craftsmanship. If you needed a chair you went to the woodworker or cabinet maker's shop and he used his tools to build you a chair. If you needed a shovel you went to the smith, who used his forge, anvil, and hammers to make you a shovel. If you needed a shirt, you went to a weaver who used his loom to make your cloth, which you then took to the seamstress or tailor who used their tools to cut and sew the garment. If you needed a violin you went to the luthier, and so on. Sure, you might also buy these items from a merchant, but ultimately they came from workers who owned the means for producing them.
In each case, economic transactions were between the consumer and the owner of the means of production, the tools, skills, experience, and so on. Sure, they might pay rent to someone else, or dues, or license fees or whatever, but they personally owed the means of manufacture. In many cases they acquired them through an apprenticeship process that both taught them their skills and provided them with the means to practice them, i.e. ownership of the means of production.
After the IR, those workers typically relocated to a factory where they ran someone else's machines-- an employment model that is so common today that we take it for granted, but which utterly destroyed a way of life that prevailed prior to the IR. Since they no longer owned those machines, which also made relatively unskilled workers better able to manufacture things that only capable craftsmen could produce previously, the only real commodity workers retained in the market was their time (and to a lesser extent their skills, but when factory owners can make workers compete with one another to work for less, skill often becomes a secondary issue-- employers can teach the limited skill set necessary to run the machines, and job insecurity helps make that worth the employer's while). To this day, most such workers are paid by the hour, and while more skilled workers might make higher hourly wages than less skilled workers, employers still pay them for the hours of their lives rather than for the goods that they produce.
Where virtually all non-agricultural manufacturing workers prior to the IR owned their means of production, most do not today. Instead, they work for someone who owns the building, the machines, the transport, whatever. The capital necessary for ownership of the means of production used to be dispersed among workers, but now it's taken almost entirely away from them and concentrated in the hands of the "owners."
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