In Aikido (and Judo, Jujitsu, etc...) we do a lot of break falling or ukemi. It is one of the first thing to is taught to beginners. It is the first thing that they are graded on and one of the hardest thing to teach and do right.
So, what are the purposes of ukemi/break falls?
You might never get into a fight, but you will fall down several times in your life.
Aside from that, if you're working in an art or practice that's going to have a lot of throwing, you need to learn breakfalls early just so you can get to the meat of your training. Avoiding breaking your wrist or collarbone is something you don't want to have to learn the hard way.
Beyond that, it's also a good cardio exercise if you're doing lots of rolls/tumbles and it's a useful warm up.
Break falling is a way to safely escape a technique that could impart serious harm to the receiver.
It is self defence at its most basic form. For obvious reasons, without it, one cannot practice Aikido safely. Thus, it is one of the first thing student should learn to do well. In no order, the purposes of ukemi are:
Apart from your own answer of safety, another practical effect is that a lot of damage in a fight isn't so much from the opponent as from the environment. Whether it's being thrown to the ground, being tripped, getting knocked back by a blow into a wall, or misstepping and running into an obstacle, that's all damage being done to you which is relatively risk-free for the attacker. To borrow a quote from Burn Notice:
In a fight, you have to be careful not to break the little bones in your hand on someone's face. That's why I like bathrooms... lots of hard surfaces. - Michael Westen
Basic ukemi teach you how to react specifically to throws, but it also teaches you the basic techniques of avoiding hurting yourself when impacting a surface, namely to spread the impact over as broad an area and distance as possible and to protect your vital spots. Tucking your head protects your head whether you're being tripped backwards onto the floor or jumping back from an attack and hitting a wall you forgot about. Training these movements over and over again make them instinctive such that you don't even have to think, but react appropriately to impact. Learning a wide variety of them makes you more likely to be able to conjure up the right reaction. Sometimes you want to roll clear of the danger. Sometimes staying more in the same place is safer (e.g., you tripped while carrying a glass and you want to avoid rolling over the broken glass in front of you or you're in a confined space where rolling will take you into a wall).
if you ever forget to hold your chin on your chest when you fall, you'll smash your head on the ground after a fall, and you'll know why you learn ukemi. It HURTS.
I've seen an olympians judoka (who have beeing intensively trained for most of his live) got to tears after failing an ukemi and hitting his head ... and when it happened, he stopped training for a while to give him time to get back together ...
If you ever fight with "normal" poeple ( as a game between friends, or in a RL situation ) you'll throw them down once ... and they'll have trouble getting up.
In Aikido, the practice of ukemi, beyond the obvious fitness' reasons, has 2 reasons:
The second reason is less obvious and more interesting:
...the second reason is the big deal, even in dojo practice. Aikido is frustrating, and spending half of the time as uke means having to deal with the fear of falling for a lot time. Overcoming the fear, leads you to overcome the frustration and to the fullness of your practice. This is not just theory, it's an empiric concept: just look at the practice of good ukes versus bad ones.
All in all, that is the essence of budo: stepping out of the mat better than you were before stepping in: with no harms, without frustration, stronger.
Don't let anybody convince you that ukemi is just a show off.
Yes, ukemi is a baseline necessity for practicing techniques, and obviously necessary to practice throwing techniques, but there are many applications past this:
Aikido ukemi practice is a crucial aspect in developing the 'soft/supple body' necessary for high-level practice/utilization. Not just how to fall safely, but how to conserve and efficiently utilize our energy. Practicing ukemi helps us learn where we carry tension in our bodies (and, to some extent, our minds), and helps us release that tension and learn new movement patterns. This knowledge of energy conservation can then bleed across into our practice and understanding of how to execute techniques.
Ukemi teaches us Aikidoka how to detect and respond to incoming threats to our physical bodies; practicing ukemi with a partner, through techniques, gives us background context to understand when a particular throw or join-lock may be imminent. This itself is a necessity for proper application of henka-waza techniques, but is also useful in practical application - physical awareness is an under-appreciated skill/sense.
Practicing ukemi teaches us how ukemi works - not only can we use our ukemi knowledge to turn a technique around into henka-waza, we can use it to understand how a person may respond to a technique and what is - or is not - too much. This in turn improves our personal practice of Aikido, our practical application of Aikido, and our teaching of Aikido.
I'm sure there are more reasons, but these are the ones I can think of that haven't been covered by other answers.
Break falls are a good way for the students to learn to practice cooperatively and safely. They let the person executing the offensive technique push through it in a way that should work against an untrained opponent unfamiliar with using the fall as an escape, so it's a useful basic fighting skill for both people. Break falls are also a form of sensitivity training: as defender, you need to learn to move in sync with the opponent - if you move too slowly you may get hurt, if you move too fast and misjudge their technique, or leave them time to change the technique on you, your attempted breakfall may go awry. The mental attitude of sensitivity is invaluable for martial arts in general - striking technique included.
Separately, breakfalls are a good way to condition the body... the impact of landing and shock through the body tones muscles big and small very effectively, as well as getting ligaments, tendons, bones etc. used to stresses. Dropping down and getting up again is useful strengthening, and it's moderately aerobic if done quickly and with intensity.
A number of techniques have the explicit purpose of crashing the attacker's head into the ground. Done well, that is the only path available for attacker. This is not the kind of technique that I would ever want to receive twice without some kind of safety built-in. A number of things get done to make sure you can practice:
