Proteins are important biomolecules that carry out various structural and enzymatic processes in cells and living things. They are also responsible for transporting other molecules in and out of cells, as well as acting as signal molecules for cell-cell communication. It makes sense that these molecules (and others such as lipids) are transported between different compartments within the cell itself, as well as from cells to their environment for the proteins to carry out their functions where they are needed.
Briefly, proteins are synthesized by ribosomes from mRNA templates and are then brought to where they are needed via protein targeting signals attached to these proteins. Proteins are brought into the nucleus and other organelles (such as the mitochondria), or remain in the cytoplasm, or are secreted out of the cell (via the secretory pathway involving the endoplasmic reticulum, the Golgi apparatus, and vesicles).
Being able to elucidate protein traffic within cells helps us understand how cellular processes work. This is very useful as it helps scientists identify what errors in the normal process are responsible for certain diseases. If proteins are not transported to their proper localizations, it could cause problems such as the loss of certain cellular functions that the protein facilitates and the excess accumulation of said protein, which could cause all sorts of complications (such is the case for Alzheimer's disease).
In terms of techniques used to study protein trafficking and localization, fluorescent tags and fluorescence microscopy are commonly used. Proteins of interests are usually exogenously expressed along with fluorescent tags (or other tags) and viewed microscopically to determine where they are localized after translation. Co-immunoprecipitation may also be used, as it determines whether a protein binds or interacts with the protein of interest. This can be used when trying to find what protein is involved in trafficking other proteins. Of course, mutation may also be used in support of co-immunoprecipitation. When a protein is mutated, it would be unable to perform its original function due to differences in amino acid composition and structure. This can confirm whether or not a suspected transport protein is indeed responsible for protein trafficking, as it cannot transport the protein once it has been mutated.