The voltage doubles because the two coaxial cables are connected in parallel on the input side, but in series on the right hand side.
In the diagram, there shouldn't be a little i in the middle of the upper coax, no common-mode current can flow there.
This trick doesn't work with batteries at DC, it just shorts out the supply, but at RF the choke, shown as the dotted lines on the upper coax, effectively disconnects it from the rest of the circuit. The coax through that choke, or wound around that core, is like a magic non-connected tunnel between left and right.
The twin-coax unun has very good high frequency performance because the two coax lines mean that the added voltages on the output are always in phase. The low frequency limit is where the choke no longer provides an impedance that is several times the output impedance, which depends on the ferrite, but within reason there is no impact on the high frequency performance by using a larger core and/or more turns. If the high frequency performance isn't important, say you only need 1.8 to 14 MHz, then the lower coax can be omitted entirely. This is also what you get from a 1:4 bifilar-wound unun on a single core.