This post focuses on the flutter kick.  The information is relevant to any swimmer, but I focus on triathlon here.  In general, I believe that the kick is the most underrated and under-trained part of swimming among triathletes, to their detriment. It is true that a strong kick is not necessary to finish a triathlon.  And in no way should this post be taken to mean that the kick is the most important part of swimming... it's not.  It would be a mistake to read this post and then train the kick just as hard as the pull or the complete stroke.  The issue is that many triathletes do almost no work on strengthening their kick.  Being a strong kicker can really benefit an athlete's swim times and can make swimming a lot easier.  Generally, in a standard non-drafting triathlon an athlete shouldn't kick very hard during the swim.  That would cause the legs to carry significant fatigue into the bike and run portions of the race for gains during the swim that are relatively small in comparison.  However that doesn’t mean that an athlete shouldn’t work to improve their kick in training. 

Increasing the power of the kick over time will allow an athlete to get more forward propulsion at the same perceived effort.  You can think of it like FTP on the bike since that is a familiar concept to many triathletes.  In most triathlons athletes race at a sub-FTP level; let’s just say 70% as an example here.  At this intensity it doesn’t feel that hard unless it is kept up for a long time. If an athlete increases their FTP by 20 watts, then 70% of that athlete's new FTP still feels exactly the same as it did before; same perceived effort, same heart rate, but power output is higher and they go faster.  Propulsion from kicking in the swim works the same way.

The kick also provides lift to the lower body.  For some who already have great body position in the water, this isn't a huge benefit.  Most could use the extra lift, though.  Without lift from the kick the hips and legs tend to sit lower in the water.  This increases frontal drag, causing the athlete to slow down.  Wetsuits basically take care of this in races where the water is cool enough. In those cases the kick is less important than in a non-wetsuit swim, though a strong kick is still helpful in wetsuit swims due to the additional propulsion and because of the third primary benefit of the kick.

Going beyond the propulsion and lift provided by the kick, a strong consistent kick also reduces the changes of speed during the stroke cycle, which saves energy.  If you ever watch someone with a slow stroke rate pull with a buoy or swim without kicking you will notice that they speed up as they pull and then slow down during the glide in between pulls.  With higher stroke rates it's less obviously noticeable, but it's still happening to some degree.  It takes a lot of effort to re-accelerate every stroke.  The laws of momentum dictate that it takes less energy to maintain a given speed evenly than to maintain that same speed by constantly slowing down and speeding up.  Your car will use less gas at a consistent 40mph than it will if it averages 40mph by constantly fluctuating between 35 and 45mph.  The propulsion provided by the kick in between pulls keeps the swimmer from slowing down as much, meaning the next pull doesn’t have to be as forceful to keep the speed up.  Less change in speed=less energy cost=more energy efficient.