I love James Bond movies also because of their opening credits.
What is the significance of the special opening credits in Bond movies?
There is an excellent article about the opening title sequences in James Bond film here -
To summarize: For the original Dr No. film, Maurice Binder, an art director at Macy's, had just received some good response to his first film title work for "The Grass Is Greener," and got a call from producers Salzman and Broccoli.
This all happened on a compressed time scale, so he essentially had to come up with a sequence he could put together in the 20 minutes he had before meeting them.
A white dot blinks across the screen, from left to right. It settles on the far right-hand edge, then opens up to reveal the inside of a sniper's gun barrel. The barrel follows the silhouette of James Bond as he walks across the white background.
Suddenly aware that he is being watched, Bond turns sharply, draws his own weapon and fires. A wash of red bleeds across the barrel and the dot falls to the bottom left of the screen, whereupon it fades away.
That was his pitch, using white circular price tag stickers to show gun shots. In the rest of the titles for that movie, dots race around the screen and circle the title texts.
The signature, racy naked girl silhouettes were not introduced until the second film, From Russia With Love.
Robert Brownjohn was a famous graphic designer in the 1950s in New York City before heroin addiction torpedoed his professional life. He heard that he could get free treatment in the UK's National Health Service (NHS), and used the boat ride there to start his cold-turkey ending of his habit. He revived his career first working for an ad agency and then forming his own, in London. He was known for being charismatic and outgoing.
Saltzman and Broccoli contacted him about doing work on the sequences, after the initial "sniper" sequence. He showed up at a theater for the interview with a carousel of 35MM slides. He dimmed the lights, took off his shirt, and did a dance, projecting the silhouette of his chubby beer belly onto the screen. The told them "It'll be just like this, except we'll use a pretty girl!"
Somewhat stunned but intrigued, they gave him a budget of just 850 pounds. He initially hired a belly dancer, but it was hard to read the titles being projected, and she left when he asked her to lift her skirt. He brought in a snake dancer (Julie Mendes) and found a way to improve the images of the text. A separate model was used for sequences where the titles were shown on the face.
The sequences were racy, sexy, and specifically labeled the movie as James Bond, and they chose to keep versions of that in subsequent movies. It was probably one of the earliest and most successful examples of branding where you don't explicitly pitch the name of the product.
EDIT: For those wondering what we're talking about, links referenced have images and videos of James Bond title sequences.
The main significance the opening credits now have are primarily just that of being part of the James Bond tradition. They are so ingrained in those films that they you can hardly take them away and in fact even the big overhauls James Bond and his films underwent over the years, most recently with Daniel Craig's debut Casino Royale, never broke with that tradtion. It is just a unique part of the James Bond trademark that adds to the building of a pop-cultural symbol immediately recognized, e.g. like the songs (which are closely related to the opening sequences and actually partly owe their significance to them), the notion of "Bond Girls" and various other things. It is just this series' unique way of structuring its credits, similar to the opening crawl of a Star Wars movie, you immediately recognize what kind of film it is (which in turn also lends itself well to parody).
If you want a more practical significance, it is just a way to structure your opening credits, which back in the 60s usually amounted to a bunch of names on a monotonous background, in an interesting and engaging way that already brings the audience into the mood of what's to come. As current title sequence designer Daniel Kleinman (who picked up the tradition after Robert Brownjohn and Maurice Binder) says in this interesting article on the matter:
"Basically, the point is to make a list of names seem inherently exciting," says Kleinman. "Traditionally, the audience have just sat through the white knuckle ride of the opening scene and a boring list of crew members could really slow down the action at that point. So I need to make something that continues the atmosphere, hints at what’s coming without giving too much away and get people in the Bond mind-set by establishing all of those familiar tropes."
In this way the opening sequences have also always tried to incorporate important themes and story elements from the movies themselves, from the voodoo elements of Live And Let Die, over the struggle of moving espionage onward from its former Cold War heritage prevalent in Goldeneye, to the gambling motif of Casino Royale, to name just a few.
The specific themes and their design have changed quite a bit over time, due to technological advances, the need for making something new every time and also the change of the Bond films and their tone themselves1, so it's not that easy to pin down specific themes that all of them transport. But there are some identifying key parts to most of them, which yet again emphasize themes closely connected to the Bond films in general, like pretty girls in suggestive poses, action, danger, Britain... All things that make the James Bond films what they are and that producers Broccoli and Saltzman wanted to transport when they first created the Eon-produced Bond series. (And while Dr. No's title sequence is rather abstract and simplistic, the more iconic themes were introduced as soon as Robert Browjohn was brought in for the 2nd film From Russia With Love.)
Saltzman had the James Bond rights, but it was Broccoli who brought with him the "formula for his past successes: exotic locations, high adventure, handsome men and beautiful women". He wanted the 007 films to have the same spirit that was captured in Hitchcock's North By Northwest ("The well-dressed and well-mannered players engaged in espionage, sexual duplicity, murder and daring acts of violence") and the World War II sabotage adventure The Guns of Navaronne.
And in addition to all that, you also have to keep the songs in mind. With the introduction of the iconic Bond theme in Dr. No's title sequence and the employment of renowned musicians for writing iconic song for each film, there has been a close connection between the opening credits and the songs. So to some degree the title sequence also is somewhat of a stage for the Bond songs, a tradition as important as the title sequences themselves. If you imagine that without an elaborate opening sequence the songs would probably play only during the end credits, as with pretty much every other movie, they would also loose much of their significance for the films and the Bond mythos in general.
1) For example it's noticable how the girls got less important during the later years, primarily the Craig years, and the danger, murder and betrayal aspects won in significance, much in congruence with the darker and more serious tone the movies themselves took, a trend only halted in the most recent and very traditional SPECTRE.