In all of these cases it's because they are Endorheic, which is a fancy way of saying that they have inputs but no outputs. The salt that is carried down by the precipitation is trapped in the lake with no where to go and so over time, the lakes become saline.
Endorheic lakes are usually in the interior of a landmass, far from an ocean in areas of relatively low rainfall. Their watersheds are often confined by natural geologic land formations such as a mountain range, cutting off water egress to the ocean. The inland water flows into dry watersheds where the water evaporates, leaving a high concentration of minerals and other inflow erosion products. Over time this input of erosion products can cause the endorheic lake to become relatively saline (a "salt lake").
Salt lakes form when the water flowing into the lake, containing salt or minerals, cannot leave because the lake is endorheic (terminal). The water then evaporates, leaving behind any dissolved salts and thus increasing its salinity, making a salt lake an excellent place for salt production.
Freshwater does contain a little salt, eroded from rocks and soil. Even though salt constantly washes into freshwater lakes, it also flows out at the same time, eventually making its way into the ocean.
The Great Salt Lake is different from other inland lakes because it has no outlets to the ocean. Three rivers empty into the lake, but the only way water can exit the lake is by evaporation. When water evaporates out of the lake, it leaves most of its salt behind, just like in the ocean. In fact, the chemistry of the Great Salt Lake is similar to the ocean, except that it’s even saltier.
The Great Salt Lake
This isn't specific to the Himalayas, there are quite a few lakes around the world that have this condition.