As detailed in the documentary "Midnight Movies: From the Margins to the Mainstream," The audience participation phenomenon grew out of the "midnight movie"/"cult" experience in the 1970s; people who identified strongly with the themes and characters came back to the show again and again. It began with a type of cosplay (dressing up as characters), then callbacks (interacting with the dialogue) and then shadowcasting (acting out the movie beneath the screen, often with interpretations).

Props were an offshoot of callbacks -- the first included the use of water pistols and newspaper to simulate a couple caught in the rain, and the sparking of lighters when the line "There's a light…" is sung. Some props have fallen out of fashion due to theatre restrictions. The throwing of toast is an example -- cleaning up bread crumbs is more difficult than sweeping up rice -- and few theaters allow open flames anymore.

A showing of The Rocky Horror Picture Show is different from everything else you have seen. If you are open minded and if you love cinema, you should watch it as soon as you have a chance. You don't need to know anything about the movie (you really shouldn't in my opinion, that's part of the magic), just don't watch it at home.

The movie was a box office failure in the U.S. in 1975. The producer, Lou Adler, decided to move it to the midnight movie circuit in 1976 to try to get some of the investment back. That was when the cult began.
At first, the movie was a success among the transvestite community and science fiction fans, but then it got out to the general public and became a national sensation. It is hard to explain why it happened, but the snowball effect is easy to understand: you are watching a movie with a crowd of people dressed up exactly like the characters, who know every single line in the script and each time there is a more questionable quote they shout back at the screen with a witty comment or a sexual joke. So it can happen pretty much anything.

I probably didn't answer your questions, but the fact is there isn't an straight answer to the why, what or how. You will understand everything after you go out and watch a night session of The Rocky Horror Picture Show.

Edit just to add a few things about the movie itself. This is a musical/comedy. It's a tribute to classical science fiction movies and, at the same time, it carries a strong message about liberty and about exploring the limits of one's sexuality. It has a brilliant soundtrack and a simple (almost naive) humor. Whether live or at home this is not a movie that everyone will enjoy. Not everyone enjoys musicals, not everyone will laugh (specially if you are watching it alone) and the sexual content may be a little too much for some people. Nevertheless, it is a unique live experience.

There is a link here that goes over all the props used during a Rocky Horror Picture Show showing. The toast, for example, is thrown when one of the characters proposes a toast (these guys are nothing if not literal).

If you want, try watching the movie at home. It's an okay movie with a HUGE amount of camp in it. However, if you go to a midnight showing with a bunch of people who act out some parts of the scenes, it becomes something more akin to a 4-D showing (so to speak). I went to a midnight showing once a LONG time ago, and they usually have 'virgins' (people who have never seen the movie before) come in front before the showing, and have some of the women try and give their best fake orgasm. Like I said, a LOT of camp...

The Most Common Elements:

"Virgin" Initiation:

If it is your first time seeing RHPS in the theater, you're a "virgin". Some theaters don't do anything about virgins, but most do. Here's what I went through as a confused, embarrassed 12-13 year old at my first RHPS showing.

• The shadowcast (see below) asks the audience if there are any virgins in the theater. The unlucky virgins are typically exposed by their friends and forced to go to the front of the theater.

• The virgins may be given some sort of token or souvenir of the event.

• The shadowcast and audience engage in a back and forth verbal ceremony:

Cast: WHAT ARE THESE PEOPLE?

Audience: VIRGINS!

Cast: WHAT ARE THEY GOING TO DO TONIGHT?

Audience: GET F#$KED!!!

(Everyone goes crazy and cheers)

I've heard of more elaborate initiations, involving sitting on a cast member's lap, being sprayed with silly string, etc. The initiation ceremony - if there is one - will vary by location.

Dressing Up:

Not everyone does it, but most hardcore fans wouldn't be caught dead at a midnight showing of RHPS in their street clothes. Dress up like a specific character; wear lingerie and face paint like the cast in the floor show scene; dress up in an outlandish and unrelated costume; wear whatever you want. The point is to have fun and let loose, and it is up to you to decide how best to accomplish this.

According to Wikipedia, attending screenings in costume began spontaneously, but was intentionally encouraged early on in the theater run:

Rocky Horror was not only found in the larger cities but throughout the United States where many attendees would get in free if they arrived in costume.... Before long, nearly every screening of the film was accompanied by a live fan cast.

Shadowcast performances:

This happens in virtually every theater that shows RHPS at midnight on Saturdays. A group of volunteers mimes the entire movie at the front of the theater as it is shown on the screen above them.

The shadowcast performances began in the Waverly Theatre in New York in 1976; the first shadowcast was led by Sal Piro and Dori Hartley.

^{Dori Parker and Sal Piro shadowcast performance, 1977}

Call-Backs/Shouting at the Screen:

What you shout will vary from place to place, but shouting is always a part of the experience. It is widely accepted for individual audience members to shout out their own witty comments as they see fit, but be warned: if you break ranks by shouting something you made up, you may be heckled and shouted down. On the other hand, if the crowd enjoys your improvisation, it may be adopted into the local repertoire.

The one universal feature of the shouting is that no matter where you are, Brad Majors (Barry Bostwick) is an "asshole", Janet Weiss (Susan Sarandon) is a "slut", and the Criminologist (Charles Gray) is "boring" and "neckless".

Audiences were singing along as early as the film's original theatrical release in 1975, but the "call-backs" (i.e., shouting at the screen) are said to have begun at the Waverly Theatre in New York on Labor Day, 1976; tradition claims that a schoolteacher named Louis Farese, Jr. was the first to engage in call-backs. His contribution came during the scene in which Janet (Susan Sarandon) is using a newspaper to keep the rain off her head; he said:

Buy an umbrella you cheap bitch!

Sal Piro urged future audiences to adopt this call-back, then incorporated others from later screenings into the repertoire. Soon, the tradition had spread to other theaters. This at least explains when people began to participate in the movie; as for why, you'd have to ask Mr. Farese.

Props:

Some of these have been phased out by certain theaters, but at one time, they were almost universal. You should find out ahead of time what is and isn't allowed at the specific theater you're attending.

Throwing Toast:

During the dinner scene:

Frank: A toast! To absent friends.

Throwing Rice:

During the wedding scene, when the cast throws rice in the movie:

Throwing Toilet Paper:

When Dr. Scott bursts through the wall, to which Brad exclaims "Great Scott!"

Because Scott is a popular brand of toilet paper in the US

Wearing Newspaper Hats and Squirting Water Pistols:

During the rainstorm scene:

Holding Up Lighters:

During the line "there's a light" in the song (There's a Light) Over at the Frankenstein Place.

Doing the Time Warp:

This should be self explanatory. I suspect that this is where the whole audience participation thing began; the criminologist actually explains how to do the dance (as do the song's lyrics).

Variations and Scripts:

I've seen RHPS maybe 10 times in the theater; I've seen it in Atlanta, Georgia; Mystic, Connecticut; Boston, Massachusetts; and New York, NY. The audience participation lines seem to evolve on their own at each theater, with some lines being ubiquitous regardless of location, and other lines being unique to a specific theater.

Some guides:

Participation: A Virgin's Guide (RHPS Official Fan Site)

Audience Participation Script (Bad Movies.org)

Audience Participation Script (Mortal Journey)

The Complete Fan Guide

RHPS Etiquette (RHPS Official Fan Site)