If it is true that a large aperture means shallow depth of field..a.k.a near objects in focus and background or far objects in blur, then how come big aperture is recommended for landscape photography?
In landscape photography, far objects like mountains need to be in focus so how a big aperture a.k.a shallow depth of field will achieve this?
You didn't explain where you read this, or what the meaning of "big aperture" means to you - so I'll explain.
Certainly you can shoot landscape photography at whatever aperture you wish. Shooting with a wide open aperture is not the most common aperture selection for most landscape photography though. By wide open, I of course mean a wide aperture such as f/3.5, f/2, f/1.4 etc. The typical reason for this is when you might want both the foreground and background in focus, which requires a larger depth of field. Depending on your focal length you may be able to still use something in this range(i.e. f/4) and still get acceptable focus throughout.
On the other end of the spectrum, you wouldn't want to shoot at the smallest aperture available on your lens, such as f/22. One reason is because of the small amount of light(along with tripod needs), but another reason is the diffraction limit of lenses. You can read about this more here: What is a "diffraction limit"?
Typically for landscape photography I'll shoot with something right in the middle(maybe f/11), that still gives me reasonable sharpness throughout the image, but doesn't hit the diffraction limit either. I also will consider the hyperfocal distance to really fine tune where I focus. See: What is "Hyperfocal Distance"?
Also see: Can I use a large aperture of f/2.8 while shooting landscape photography?
Most of the time, a "big" f-number is recommended for landscape photography. But a high f-number such as f/16 or f/22, when dealing with APS-C or FF cameras, means a very narrow or small aperture. The large aperture is at the other end of the scale at f/1.4 or f/2. See What is aperture, and how does it affect my photographs?
I'm not so sure that landscapes are usually taken with "big" apertures. However, there is a reason large apertures make sense in some situations.
When there is nothing close to the camera in the picture, as can be the case with landscape shots, most of the scene will be at infinity focus. In that case there is no benefit from a larger depth of field, so you can use the sharpest setting for whatever lens you have. This is usually a couple of stops or so down from wide open.
Diffraction goes up with smaller aperture, but lenses aren't perfect, and wide open will be a compromise between different parts of the design. Usually a couple of stops down from wide open reduces abberations and other quality problems with the lens full open, but is still a wide enough aperture that loss of sharpness due to diffraction is below other sharpness limits.
This will be different for very high quality, and therefore expensive, lenses. Those will be better relatively at wide open, so you want to stop down less for the best sharpness.
