This is so simple that I have not bothered to make a schematic. All you need to know is in the eevblog thread that is given as link in the video - or just search : "Easy DIY 5.5 Digit DVM + Volt Ref./Cal. (LTC2400+LTC6655 / SPI uC / Arduino)" from any search engine on the web.
This (the photo) is the minimum setup from the datasheet (linked in the thread on eevblog given over). Forget the Schmitt trigger - you don't need it. It's just SPI - plug in to digital pin 10,12 and 13. The jumpers are not needed - it is for a demo board from LT.
All you need of parts is the 8 pin ADC LTC2400 (around USD10) and a good voltage reference (I use LTC6655, but this schematics uses LT1236 - both are less the 10 bucks). And some wires and capacitors. I have made 2 versions with 1602 and 5110 display. Code is floating around the web. The datasheet, code links and other stuff are all in the link in the video description (the search link given in the beginning of this post).