You are not missing anything. Just a bit of theory has gone, or never been.
The link you gave only gave output specs for the transformer.
These are only rough numbers, but it will help you understand.
The next part assumes the output of the transformer is loaded to 800 watts.
For 230v and 800w (Ignoring efficiency and resistive loss fudges by transformer designer)
230 volts x 3.48 amps = 800 watts.
for 115v applied
115 volts x 6.96 amps = 800 watts.
If you have 2 115v windings in series with 3.48 amps x (115 + 115) = 800w.
now if you put the 115v windings in parallel and apply 115v
The windings will be rated at about 3.5 amps.
When you have 2 in parallel, the total current at 115v will be 3.48 + 3.48 = 6.96 amps.
Now for some LIVE TESTING. No fingers connected at any time!
Now lets assume there is no load on the transformer secondary.
It is very important that the windings are phased correctly.
When in series, then end of the one winding MUST go to the start of the other winding.
When in parallel the starts are connected together, and the finishes are connected together.
If you have a spec sheet which clearly says which wires to parallel for 115v or which wires to connect in series then follow the instructions.
If the spec sheet shows a 230v connection, and shows the centre series connection, then one of those wires is a start of a winding, and the second is the end of the other winding.
If you don't know which are the starts, and finishes, then you need carry out some tests.
In the following S1 and F1 are winding 1 start and finish. S2 And F2 are winding 2.
On 115v it is easy to test if the phasing is correct.
1). If we don't know the start lets take one wire of each of the windings, and assume they are starts.
2). Connect them together. S1/S2
3). Connect 115v to S1/S2 and F1.
4). Measure between F1 and F2 with a meter on ACv (> 200v).
You should get very close to zero volts. If so go to step 6)
5). If you get 230 volts then the assumption at 1) was WRONG. It was actually S1/F2.
TURN OFF POWER
If you get to here a second time CALL IN AN EXPERT. You will kill yourself.
Make new assuption for S2, and go back to step 1)
6). TURN OFF POWER.
Connect F1 to F2, and apply power. Measure outputs. You should get 60V on each of the output windings.