To accurately answer the question, "What is the purpose of martial art forms;?" you have to answer first, ...what is the purpose of martial arts? Since karate is the most popular martial art on a worldwide basis, I'll use karate and karate kata (forms) as the subject.
Karate develops the human potential on three dimensions, body, mind & spirit. The development of this potential can then be used in a fighting or self defense application.
I'm going to concentrate on the first two dimensions of human potential, body & mind, because although the mental dimension of karate is often spoken to, it is typically misunderstood and neglected.... The 1st set of principles is then development of body & mind.
In traditional karate, the standard curriculum is comprised of three main components: (1) Kihon (basics), (2) Kata (predefined forms or patterns), and (3) Kumite (sparring or actual fighting exercises). Thus the 2nd set of principles is the three-sided karate curriculum.
Only karate, as opposed to the sports-based fighting methods such as boxing, muay thai, mma striking, bjj, has the second component of martial training, i.e. kata. Thus kata training becomes a focal point on how to train for martial arts....
Returning to the 1st set of principles, body and mind, I submit the overriding difference between the practice of karate and the practice of the sport-based fighting methods listed above, is the mental dimension. Both karate and the sports-based methods such as boxing stress physical conditioning. The huge difference in karate is the emphasis on mental discipline. Under traditional karate, mental discipline is the key; mental discipline directs every movement and execution of technique, from basics, to forms, to fighting application.
Once the body is physically conditioned, the mental discipline of karate develops fighting skill in two critical dimensions. One, through the integration of the whole body's strength in a highly coordinated manner. Second, that whole body strength is used in a completely controlled, deliberate and precise way against the actions of the opponent. Highly developed mental discipline guides every single one of your actions against what ever the opponent puts forth.
It's true that practicing basics (kihon) and kumite (sparring) exercises can produce the karate result I have defined above. However, it is kata that develops the mental control and discipline over the body's coordinated working in accomplishing deliberate, powerful actions or movements---to it's highest level.
This is why solo kata practice or 'punching air' is not a useless activity, but an extremely sophisticated method of developing mind / unity and the accompanying powerful technique and effectiveness kata practice creates....
I'll stop here, surely there will be questions....
Edit (reply/rebuttal to Mark's answer)
Is kata superfluous in accomplishing excellent karate. The answer goes back to the principles I stated underlying traditional karate as a martial art.
The critical answer lies in how the mind is engaged in the training. The secret of successful karate comes from integrating the mind and body. This is problematic to demonstrate over the internet.
Let me try to illustrate that repetitively drilling physical techniques until they are imbedded in "muscle memory," is not karate mental disicpline. This kind of training produces the ability to replicate actions and creates quick reactions. We see this kind of training output in every sports activity from gymnastics to 'karate kata performances' to basketball.
Kata, or any aspect of karate, practiced this way, or performed this way is hollow and weak. The reason is the deliberate control of a highly disciplined mind has taken a back seat to the regurgitation of memorized physical actions. This is why traditional karate fails, kata fails when is comes to actual application. The fault is not the kata or karate exercises, it's the fault of the practitioner to understand how the mind is driving the physical actions.
Note that for all the commentators here, not one has broken out the fundamental aspects of a basic kata, nor described their purpose and integration.... It's not wonder kata fails when practitioners are merely performing the physical monkey see-monkey do of an instructor....
First, I want to say that a number of the posters have touched on what I'm working to describe, and have addressed the subject of Mental Discipline in certain areas.
Go to the specific, take a look at the first Shotokan kata, Taikyoku Shodan. How do we begin this kata? How is it shaped & many steps does it have? What is the significance of the ready position at the beginning and end? What the precise moves in the ready position--and what is the body, mind, and spirit (metaphysical I know) doing with the ready position? What are the steps in the 1st branch or leg of this kata and again what is the body, mind, and spirit doing at each step?
Many criticize the Taikyoku kata as simplistic and impractical for fighting, that it's watered down. Is this really true for the purposes of kata training, and why did the Okinawan masters create these?
edited for typos...
@ Sarathrion: My invitation to describe the functions in the first Shotokan kata, Taikyoku was met with, "confusing, "mythical mumbo jumbo, etc.," and a "downgrade." The real problem is the failure of "martial arts" practitoners to methodically study & examine the purpose & structure of kata. Instead, posters want to talk about their impressions of kata.... without serious challenge from someone of my point of view....