Understanding the geometrical interpretation of word2vec I'm trying to understand how the $word2vec$ method actually nudges word vectors of similar semantic/syntactic content closer together in the word vector space.

I've read here (Quora answer) that it's because $word2vec$ tries to maximise the probability of target word $w$ given context $c$, and that this probability is roughly inversely proportional to the distance between $w$ and $c$ in the word vector space. I don't really see why the latter is the case though.
If two words tend to appear in the same context, then they will tend to receive a similar word vector.

For instance, a large corpus might contain sentences like "I really love Fuji apples" and "I really love Gala apples".  From this we can see that "Fuji" and "Gala" have appeared with the same context (in this case the context is "love _ apples", if by context we look one word to the left and one word to the right).  This will tend to cause the optimization algorithm to assign Fuji and Gala vectors that are similar.

As you can see, words that tend to be associated with the same surrounding context will often have a similar semantic content.  Not always, but it's a good heuristic.