The importance of training the falsetto register has been vastly misunderstood by many singers and teachers. One of the greatest problems with discussing the falsetto register is having a proper example of what the falsetto actually sounds like. I have had many singers come into the studio who make completely different sounds when attempting to sing in falsetto. From an airy, woolly sound to a constricted mixed chest sound. The following is an accurate example of pure falsetto, sung on an OO or "U" vowel. Notice that it is dark, without many overtones, and it is also not airy.
Now, it is important to remember that the falsetto register is used as a training tool. It is not meant to be sung on stage except for comedic effect. There are many important reasons for developing the falsetto register. Before we discuss these reasons let me first give the definition of the falsetto register. The falsetto is engaged when only the edges of the vocal folds vibrate while the main body of the vocal folds- made up of the thyroarytenoid muscles - are not engaged. The thyroaretenoid muscles are the muscles responsible for the lower register or "chest voice" sound. So they are totally inactive in falsetto register.
The fact that the lower register does not participate in the falsetto register results in several significant characteristics. First, since the thyroarytenoids (lower register) are responsible for supplying the acoustical energy heard as overtones, the falsetto - lacking lower register - has a very limited number of overtones. There can only be one or two overtones present in pure falsetto. As a result, the falsetto register lacks the amount of overtones needed to sing the vowels EH,OH or AH because all of these vowels require more than two overtones to be clearly distinguished. Therefore, pure falsetto can only be sung on the vowel's OO or EE as those vowels only need two overtones to be understood. Why is this important? It is important for the training of the falsetto register. The best way to train a group of muscles, such as the falsetto muscles, is to isolate them and develop them in their pure state. So if a teacher or student is trying to develop the falsetto register by singing on the vowels "OH", "AH" or "EH" they are doing it incorrectly as those vowels must bring in some lower register to be distinguished. They would only be truly strengthening and purifying the falsetto register if singing it on OO or EE.
Once the falsetto register has been isolated it is important to strengthen it for the following reasons. First and foremost, falsetto gives the fullest lengthening to the vocal folds. Lower register uses the thyroarytenoids which, when tensed, shorten the vocal folds. The falsetto register does the opposite in that it lengthens the vocal folds allowing for the changes in pitch to be accomplished by the use of the cricothyroid muscles. As we all know, a longer string playing the same pitch as a shorter string produces more sound because it has more surface area and moves more air. Not only that, but the falsetto lengthening of the vocal folds also allows for the glottal space to be more open due to this lengthening of the vocal folds without the shortening affect of the thyroarytenoid. This greatly helps to alleviate excess constriction in the voice.
Secondly, the falsetto register supports the correct setup of the vocal cartilages for the proper stringing and adduction - approximation - of the vocal folds. The vocal folds are attached to the arytenoid cartilages - two triangular shaped cartilages - in the back of the glottal space and also to the thyroid cartilage (Adam's apple) in the front of the glottal space. So, if you are looking through the top of your head down through your windpipe, the vocal folds sit on top of the windpipe and can close over it. Also, the whole glottis can close over the windpipe like when you swallow.
Where the vocal folds come together is where your Adam's apple is and where they go back and apart is where they attach to the arytenoid cartilages. The arytenoid cartilages must be stabilized so that the vocal folds can be stretched at the front (where they come together) for the changing of the pitch. If you can imagine trying to stretch the vocal folds to change pitch while the arytenoid cartilages - which they are attached to in the back - are not holding, it doesn't make for a very good setup. And many vocal issues stem from this very problem. So the falsetto is a very good tool to get the proper holding of the arytenoid cartilages. Additionally, the arytenoid cartilages are brought together by the transverse arytenoid muscle and the oblique arytenoids. These muscles adduct the vocal folds in a proper technique. Many schools of vocal technique today teach that the Lateral Crico-arytenoids adduct the folds, and while it is true that they can bring the vocal folds together, it is far from ideal. The lateral cricoarytneoids are attached to the arytenoid cartilages. Likewise the posterior cricoarytenoids are also attached to the arytenoid cartilages. However, the lateral cricoarytneoids and posterior cricoarytenoids are antagonistic to each other. So they hold against each other in order to stabilize the arytenoid cartilages for the stringing of the vocal folds. If one uses the lateral cricoarytenoids to adduct the folds they have destroyed this vital balance. This can lead from anything from constriction to aspirated singing.
In reference to the above explanation, there is an additional benefit to developing the falsetto. When you stabilize the arytneoid cartilages it correctly causes the singer to have to use the ideal muscles of approximation to keep the vocal folds closed. Thus, the falsetto - when sung firmly and clearly - works these muscles of approximation. If falsetto is sung with an airy tone it does not work these muscles. That is why it is incorrect to sing falsetto airy. So with the lower register out of the way, the proper stringing and adduction of the vocal folds is allowed to take place while the cricothyroid muscles stretch the vocal folds.
Lastly, the cricothyroids are also important as they come more and more into play as the singer goes higher in their range. If they are weak, the high notes are nearly impossible to sing. When the proper setup of the falsetto is established, and the cricothyroids are strong, proper lower register can then be added to this setup. And that gives the most ease, beauty, power and range. Dynamic singing is all but impossible without a greatly developed falsetto as the singer will tend to constrict the glottal space to lessen the sound. Instead, in a good technique, the singer lessens the participation of lower register (the thyroarytenoid) while keeping the glottal space open. This brings the sound more towards the falsetto, while just keeping enough lower register participation for clarity. If the falsetto is not strong then, this maneuver is impossible to perform correctly.