Replacing old 3-switch with new one that has Line In
Fixing the sins of installers past

What the installer who put this switch in did is run two /2 NM cables between the switch and the junction box downstream of it to hold all the switched hots and the neutral, which was rather naughty of them as it is a likely NEC 300.3(B)/310.10(H) violation that could get your switch box mighty toasty if the circuits the triple-switch was controlling were heavily loaded.

What they should have done, and what your electrician (as you indicated you weren't comfortable running wire yourself) needs to do to fix it, is run a /4 W/G NM cable from the switch box to the first downstream junction box -- that cable has enough wires in it for the neutral, the ground and all the switched hots, combined into one sheath.  Then, they can put the new switch in and have the installation (or at least, this part of it) meet Code :)