A distro being ‘stable’ generally means that there will be no breaking changes in the API/ABI it provides over the course of the lifecycle of a given release.
This means, in particular, that software written to run on a given release of the distro will (mostly) not need any changes to keep working correctly on that same release of that same distro. This guarantee is exceedingly important for enterprise and infrastructure usage, because it means you can keep your underlying platform up to date in terms of security without having to put a lot of effort in to keep your own applications and services working correctly on it.
This concept can actually go beyond the basic idea of a given release, and extend to major aspects of the distro design itself. This is actually why Ubuntu is often called less stable than Debian even though they both adhere to the above definition of stability, Canonical (the compan behind Ubuntu) has developed a reputation for making big, highly visible, changes to how the distro works which require significant changes not only to software running on it but also how users interact with the system (the first big examples were Unity and Upstart, the most recent one is their push to force everyone to use Snap packages instead pf APT).
Now, in terms of end-user impact, unless you need that comparability guarantee or have an atypical setup, there’s not much impact for most people in terms of using an ‘unstable’ distro other than needing to update more frequently to stay secure.
In fact, there’s a major benefit to using ‘unstable’ distros in that you will almost always get new versions of software much more quickly than you otherwise would (Debian’s stable releases are an extreme example of this working in reverse, many things in Debian 10 are months or even years behind the upstream projects). This goes double for trying to use brand-new hardware, as many ‘stable’ distros use old kernels that lack support for the newest hardware.
As a side note, different distros may have differing manings for the terms ‘stable’ and ‘unstable’.
In the case of Debian, ‘unstable’ is a compilation of the most recently packaged versions of all packages available in Debian. It serves as a base for their ‘testing’ version, which is what will be the next release of Debian (but is usually behind ‘unstable’), and is actually perfectly usable for general purpose daily usage by itself, while it’s the ‘experimental’ repositories that are truly dangerous territory (they contain things that the distro maintainers have not tested enough to be certain they play nice with the rest of the system).
In contrast, with Gentoo (one of the more popular rolling-release distros), the difference between ‘stable’ and ‘unstable’ is simply a matter of how rigorous the testing of that particular version of that package has been. Even there though, ‘unstable’ is not particularly dangerous most of the time provided you actually know what you’re doing (I’ve been using Gentoo unstable for about a decade now, and have only had one time I had to rebuild a broken system because of it).
Most other distros fall somewhere in-between, or actively choose to avoid using such terminology, instead referring to ‘releases’ (which are implicitly stable) and ‘development’, ‘next’, or something similar, with the distinction being mostly on-par with the difference between Debian’s ‘stable’ and ‘unstable’.