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'...under the impression that with a twisted pair connection, there is no voltage across that pair'
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There is a nominal 50V d.c. voltage on the line. When the handset is lifted, a loop is made and the current operates a relay at the exchange. A d.c. excitation current is then applied to the subs' telephone to energise the carbon mic (used to good effect by memory storage and, in some cases, speaker amplification and the illumination of an LED). The line-length may present a resistance of 1 kilohm or so.
The audio tones and speech are superimposed upon the d.c. excitation voltage and bridged off through capacitors at the exchange, with the excitation voltage fed to the telephone via a bridging coil (relay) steering the a.c. speech and tones one way and the d.c. another.
The above is very simplistic!