Do I have to watch all eight movies in order to fully understand the significant plot points found in 'Fantastic Beasts and Where to Find Them'?
No. The Harry Potter films arguably followed the books for the most part, and none of the significant elements in Fantastic Beasts were based on the movie inventions. Just reading the seven novels should be enough, though you may also want to read the three tie-in writings that Rowling released online. (1, 2, 3)
Is there anything that I'll miss by not having seen them?
There are several visual elements that take reference from previous Harry Potter films, due to the fact that MinaLima is still doing the artwork and David Yates is still directing. None of these are important to the plot. The only parts I can think of which may particularly jarring to someone who hasn't seen the previous movies is that there is an unexplained instance of priori incantatem (which in the books only happens when wands have twin cores) and there is an instance of a person using magic to fly (which in the books is generally said to be impossible).