Q:

How do the windings in a residential transformer create a neutral and why are the hots opposite phase? 

Pretty self-explanatory I think; just wondering why neutral "manifests" from the windings and why it "outputs" (for lack of a better word) a negative-phase hot?   I guess my perplexity comes from thinking that the [+hot/neutralwindings/-hot] are all the same wire so assuming that is correct what causes the changes? 

A:

The transformer doesn't manifest neutral: the ground rod does. 

Or at least it should, according to electric codes.  The "neutral" wiring is connected to the "safety ground" at exactly one point, and the safety ground is connected to this big copper rod stuck in the Earth.  It's "neutral" because it's connected to Earth, and since you are probably also touching (perhaps indirectly) Earth, neutral won't shock you (absent any wiring faults, which is why there's a separate safety ground, which should be the same as neutral). 
As a schematic, it looks like this:

simulate this circuit – Schematic created using CircuitLab
Because the "neutral" is the center tap of the transformer, at any point the voltage at neutral will be exactly halfway between the voltages of the "hots".  Thus, V1 is 120V, V2 is 120V, and if you add the two together you get 240V at V3.  Or put another way, V1 = V2, both in terms of AC, or DC voltages at any instant. 
The reason they are in opposite phase should be evident from the schematic: V1 has neutral on the "-" side, while V2 has it on the "+" side.  As I've drawn the schematic, V1 and V2 are actually in phase, but if you flipped V2 around so that the "-" was on neutral, then they would be opposite phase.  Because this is AC, which side is neutral and which is hot is largely irrelevant, except when you consider minor details like safety. 

A:

The transformer secondary consists of a really long wire.   The 'hots' are connected to the ends of the wire, and the 'neutral' is connected to the middle.   The wire is then wrapped into a coil wound around the transformer core.   Current flowing through the primary side of the transformer creates an oscillating magnetic field in the transformer core.   This field induces an electric field in the secondary.   Since the whole winding experiences the same magnetic field (more or less), the voltage induced will be the same along the entire coil.   If at one point in time 100V is induced along the entire coil and the 'neutral' tap is exactly in the middle, then you will see 50V between the 'neutral' and the 'hots'.   Now, if you consider the 'neutral' to be 'zero potential', then one 'hot' will be at -50V and the other will be at +50V.  You can just as easily consider one 'hot' to be at zero potential.   In this case, one 'hot' will be at 0V, the 'neutral' at 50V, and the other 'hot' at 100V.  The 'neutral' connection is really a matter of perspective.   This relationship holds with any signal passing through the transformer.   If you put an AC signal through and consider the 'neutral' to be 0V, then the two 'hots' will be inverted copies of each other.   If you consider one of the 'hots' to be zero, then you will get two waveforms with the same phase out of the 'neutral' and the other 'hot', but the waveform coming out of the 'neutral' will have half of the amplitude. 