Process nodes are about how small the transistors (or other measurable features) can be made. Everyone uses a different measurement criteria, so the numbers aren't 100% comparable between companies.
For the manufacturers, it means more profit. Smaller chips equals more chips per wafer, and as long as yields are good, more money on sales. (Intel is having yield problems with their new 10nm process, If I recall they aimed for a 28% reduction in size, which is a bigger jump then they have made before)
When you make the transistors effectively smaller, it takes less voltage to flip their state. Lower voltage does mean lower power consumption, and this typically means that on high end parts, high clock speeds can be achieved for a given temperature envelope. You do run into some complications with capacitance, and they are just about at the limits of what traditional lithography can achieve in terms of size.
More cores is just a matter of using more silicon, but yes, since they run at a lower voltage you can get more cores into things like laptops.
TSMC's 7nm isn't quite as small as Intel's 10nm. But basically if AMD can get that to market first it will seem like a huge advantage. Their CPUs on the 14nm and 12nm processes are competitive already. That next jump will make a pretty big difference.
Intel already has 10nm chips in production, just that the yields are low, so they are probably losing money. Very likely why they haven't gone for mass adoption and keep improving their 14nm lineup.