More VRAM allows your computer to load more and higher resolution textures or 3D meshes onto your GPU, as well as render images at higher resolutions (multiple monitors is the same as having a higher resolution).
When your GPU does not feature enough VRAM, it will load its ressources onto the system RAM instead. However, due to the system RAM's distance to the GPU, it is a lot slower than VRAM.
If your GPU's VRAM is full during a game, chances are that you might see your FPS boosted a lot by upgrading your GPU's RAM (which is unfortunately not possible, since VRAM is soldered onto the graphics card)
This is partly the reason, why IGPs (Integrated Graphics Processors) are much slower than dedicated GPUs, since they have no VRAM and thus have to rely on the slower system RAM.
Modern games nowadays try to use as much VRAM as is available.
Sometimes in an effort to reduce pop-ins by pre-loading assets that aren't visible on screen yet, sometimes in order to improve visual quality by increasing the number of foliage or other similarly decorative objects. This can lead to the VRAM being completely full at all times without necessarily impacting performance.
Because of that, even if your VRAM is full, there is no need to panic. You only need more VRAM if it is full, your GPU is mostly idle, and your framerate is bad.