Basically the idea behind radiosity is, you divide the surfaces in your scene into discrete patches (i.e. tiny rectangles mapped onto everything). A patch receives energy from a light source, or reflected energy from other patches, and it reflects all of that back into the scene (conservation of energy).
The assumption is, that a patch is perfectly diffuse and it reflects in all directions equally. Imagine a fish-eye-lens perspective (actually a hemisphere) from a patch into the world. The projected size and visibility of all the other patches gives you the proportion of energy transferred to/from that other patch.
> Was it a performance matter ...
Computing the final color of a patch is somewhat computationally expensive. It involves either inverting a giant matrix, or iterating until you reach a stable result. The end result is not a rendered image, it's a perspective independent light distribution in your scene. You still need to combine that with another rendering technique, like ray-tracing or rasterization to actually get a picture.
Computer games of the era you mention (e.g. the Half-Life series and other GoldSrc/Source engine based games), did radiosity in an offline, pre-computation step and stored the patch colors in a texture (this is usually called a light map). Using the light map at runtime is pretty cheap.
> ... or was it a physical model limitation (you can't simulate all light phenomenons) ?
As I described above, radiosity assumes perfectly diffuse reflectors. It can't do mirror-like specular reflection of light. Ray-tracing and radiosity actually complement each other really nicely in that aspect, as mirror like reflections are trivial to do with ray-tracing and you can integrate light maps for indirect, diffuse bounces.
That said, with modern hardware, doing radiosity in real-time even for reasonably sized scenes has become feasible. There is e.g. the proprietary "Enlighten" engine that claims to do real-time radiosity. There are other approaches that have become feasible as well, like doing real-time photon mapping. Ray tracing hardware finally being a thing is also certainly a game changer. Well over a decade ago, Crytek did some work on voxel grid diffusion based stuff and Nvidia worked on integrating cone-tracing into that. I have little idea tough what modern game rendering pipelines look like, I've been pretty much out of touch with that industry for quite a while now.