Even a small amount of weight in a backpack can cause loaded walking. This means the compressive force on the spine is enough to increase the load on the discs. The discs are fluid-filled to help cushion the spine. Any pressure against them can cause small amounts of the fluid to leak out.
Narrowing the disc space even a little puts more pressure on the joints of the spine. The back muscles respond by contracting to help support the spine. All these factors together can add up to pain or discomfort for some people.
Why are some people affected and others aren't? There's no clear answer to that question. It may have to do with individual anatomy. Perhaps some people have smaller disc spaces to begin with and any change causes an immediate problem. Maybe muscle activation is faster in some individuals causing increased compressive forces sooner than in other individuals.
A recent study of back pain patients wearing a weighted vest while walking showed increased pain and a longer recovery time afterward. It took longer for the muscles to relax and the disc spaces to return to a normal height.