Salt can affect your landscaping in several ways. The first is how your yard will be affected. Salt will become a liquid as snow melts, and all of that liquid is absorbed into the ground. When the salt is liquefied, it separates into sodium and chloride, and your grass and plants will absorb this and leave less room for them to absorb the nutrients they need. This will interrupt the photosynthesis process, which will lead to leaf burn, when they turn brown, and will kill grass and some parts of plants.
There is also the salt spray on plants near the road. The salt sitting on the plant will cause salt burn on buds, small twigs, and leaves. The salt can also burn away the outer layer of buds, exposing them to harsh winter winds which will kill the buds. Most of the time damage will not be shown until late winter/spring, when the plant will not start to grow back, or needles or leaves on trees will turn brown.
Aside from dissolved salt being absorbed into plants, the salt that is in the ground could also keep plants from getting the water they need. Rock salt will absorb water and keep grass from getting the water it needs, which will stunt growth. This process is not so quick acting, you may not notice this until late summer or even years later. One way to avoid this is to aerate your lawn.
There are other factors to how plants are affected by salt. If there is proper drainage that might carry some of the salt away. Salt applied later in winter will also cause more damage than salt applied earlier in winter, as late winter salt will be more directly affecting the water consumption of plants, and the plants may absorb more of the late winter salt when they come out of dormancy. Rain can also help wash some of the salt off of leaves and prevent salt burn.