The (likely) origins of this trope
There is only one place in The Bible where Satan is given a physical description:- "And there appeared another wonder in heaven; and behold a great red dragon, having seven heads and ten horns, and seven crowns upon his heads ...... And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world" - Revelation 12:3, 9
So that's probably where the "red" component of the Big Red Devil comes from. If you were wondering why depicting the devil as a giant dragon isn't so common, well 1) it was somewhat more common in medieval Europe, and 2) this scene in the Bible is generally accepted to be the devil going One-Winged Angel mode and not his "normal" state.
But what about the goat-like features - the legs, the horns, etc? There are two possible sources, both of which probably reinforced each other. The first one is Jesus's parable of sheep and goats, where sheep represent those who belong to Christ and goats those who do not (as Jesus is analogized to a shepherd). The second one is the Demonization of the ancient deity Pan from Classical Mythology - the early church understood demons to be the forces behind pagan idols, and Pan was probably focused on for being less humanoid than the others. Fauns and Satyrs were often (but not always) associated with the devil in iconography.
It should be noted that this depiction wasn't always the most common one. In Islamic and Eastern Christian art, it was always a Big Black Devil. Medieval Christian Western Europe usually opted for either a creature with the traits of many different beasts and with faces where faces shouldn't be, or put him in all black like their cousins in the east did. Certain animals, especially snakes, were often used as a shorthand for the devil in all traditions. That's not to say that they never did a Big Red Devil, just that it wasn't nearly as iconic back then. Further reading HERE
. The common image of the well-dressed man with horns and perhaps red skin comes from Goethe's Faust (with a hint of Biblical influence, as scripture describes the devil as beautiful). As noted on the main page, he dresses up in red in Faust, which probably helped to popularize the trope.
