A 4K texture, does not require a 4K monitor.
In short, the graphics pipeline for getting "pretty pictures" on a screen is, with some hand-waving to simplify things:
You have a 3D model, that looks like a thing - e.g. "a tree"
You have a texture, which is just a 2D image of colors
The texture is mapped to the model, such that each point on the surface of the 3D model (split into triangles), has a co-ordinate on the texture. i.e. points to a specific pixel on the texture. It's like mathematically wrapping a candy-bar wrapper around the chocolate inside.
When you view this in game, Maths™ is used, which maps each point on the 3D model, to a pixel on your screen.
As each point on the 3D model, is already mapped to a pixel in the texture. You can now set the color of that pixel on your monitor - to the color from the texture.
Note, the obvious things now:
There are infinite points on a 3D model (look at your hand, find two points - you can keep finding a "half-way" point mathematically, infinite times)
There are a limited number of pixels in the texture map. It has a width and height, and there is clearly no such thing as half-way between two pixels - it's either pixel 1, or pixel 2.
Therefore, to map each point on the model to a pixel in the texture, means some points will share a pixel. In game, when you zoom too far in - this means you either get pixelated looking images, or if they use any kind of filtering (*all modern games do), you'll get a nice big blurry spot.
A 4K texture just has more pixels to sample from. So more points on the model will have their own unique pixel. That is, you could get much closer to it - and you will still see details.
The resolution of your monitor only matters after this point. The model will still be rendered using the high-quality textures, even if your monitor is so bad - that you'd have to stand right next to the model to see any difference.