These lenses work basically like a magnifying glass (or reading glasses) in front of the lens. In fact, not just basically — actually exactly.

That has the effect of decreasing the distance from the back of the lens to where an in-focus image is formed, which gives the lens more freedom to focus more closely. And focusing more closely is inherently all macro lenses do. They let you focus on objects very close to the camera relative to the angle of view, which means you can make the object fill more of the frame.

So, how do add-on macro lenses do this? Well, normally, a lens focuses at infinity (see What is "infinity focus"?) at its closest designed distance from the sensor*, and the camera focuses on closer objects by increasing that distance by the right amount. But there's a limit to how close that can go — given the physical constraints of the lens construction, they can only go so far — leading to the minimum focus distance. In a smartphone camera, there's not much room at all, so a relatively high minimum focus distance is common.

The close-up supplementary lens changes the light path so that the in-focus image is formed closer to the back of the lens (and further from the sensor). It actually decreases the effective combined focal length, without changing the distance the lens is placed from the sensor.** That means everything is inherently focused closer — making that physical limit correspond to a much closer minimum focus. But nothing comes for free, because at the other limit, where the lens used to focus to infinity, it now can't get close enough — just like wearing reading glasses makes it hard to see in the distance.

(This is the same effect you'd get by spacing the lens further from the sensor with extension tubes — the lens's "native" point for forming an image is now shifted further away from the sensor, leaving more room for close focusing and less for distance focusing.)

And to continue with the nothing-comes-for-free theme, because these lenses are usually cheap single-element lenses or at best doublets, they usually degrade technical image quality — reduced sharpness, increased chromatic aberration. Of course, on a smartphone, it's not like you have other options, but you're unlikely to get results as good as an interchangeable lens camera with a dedicated macro lens, or with extension tubes (since those do the same thing without requiring more glass) — or even a point & shoot which can move the lens to focus at macro distances natively.

Note that this is the same for supplemental close-up lenses (or "macro filters", or whatever they may be called) for all cameras, not just smartphones.

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* technical caveat: modern lenses actually usually allow some play beyond infinity, but that's to make it easier to actually get infinity right, not because that's useful for anything.

** It's sometimes repeated that these macro lenses don't change the focal length. That's a myth. There's some math here, but I'm going to make it a different Q&A, since it's really tangential to the basic question here.