Shrinking a process gives several benefits, but a quick general overview helps:
Silicon as used in chip manufacturing is expensive. It costs a lot to grow, cut and polish. It's also a mature industry, so no real breakthroughs are likely to happen to reduce the cost of the silicon. The less silicon area you use, the more chips you can make for the same cost. Next is manufacturing. Whether you put one transistor per square millimeter or 100,000 per square millimeter, the cost is the same, or at least within a penny. Coat, expose to a masked pattern, etch, sputter, clean and repeat a few times, and voila, you have a chip. Shining a light through a mask costs the same no matter the resolution of the mask. Dunking the wafer in a chemical etch bath is the same, running a wafer through a sputterer or CVD costs the same, etc. Labor costs are basically per wafer, so more components per wafer means you get more output for the same labor (and plant infrastructure) dollar.
So, a smaller manufacturing process means:
More components per wafer. Thus if you double the component density, your manufacturing costs will remain the same, and you can double output while keeping costs the same (think 32GB for the price of 16GB).
You can also make the chips smaller while keeping the same capacity (same 16GB chip uses half the silicon, thus costs 50% less to make, think 16GB for half the cost you paid last year).
Or, more capacity within given size limits. (think 64GB or 128GB SD cards, or 2 TB Compact Flash).