Like any trained athlete, Captain America will tell you he is only as good as the effort he pushes his body to perform.
The Super-Soldier serum did make his body physically fit, but being at the peak of human capacity and being battle-tested are entirely different things.
While we are never told if Captain America needs to exercise to maintain his beef-cake physique, in the comics, the Captain is always training to increase his combat effectiveness, his battle awareness, his strategic sense of the battlefield and working to decrease his reaction time.
Steve Rogers' physical transformation, from a reprint of Captain America Comics #1 (March 1941).
Art by Joe Simon and Jack Kirby.
While we have seen the Captain go for months without training to no ill effect, it would seem the more he trains, the more confident he feels about his abilities under fire.
Given he is one of the members of the team with comparatively limited superhuman abilities, he may feel an extra incentive to push himself to his physical limits.
Early comics confirm this perspective.
Captain America may also train because he can do things ordinary humans cannot easily do and the only way to reach those peaks of performance and enter the flow state where they become effortless is to practice, practice, practice.
Captain America has no superhuman powers, but through the Super-Soldier Serum and "Vita-Ray" treatment, he is transformed and his strength, endurance, agility, speed, reflexes, durability, and healing are at the zenith of natural human potential.
Rogers' body regularly replenishes the super-soldier serum; it does not wear off.
The formula enhances all of his metabolic functions and prevents the build-up of fatigue poisons in his muscles, giving him endurance far in excess of an ordinary human being.
This accounts for many of his extraordinary feats, including bench pressing 1200 pounds (545 kg) and running a mile (1.6 km) in 73 seconds (49 mph/78 kph).
REF:
Captain America's Powers and Abilities; Wikipedia.
In the end, Captain America may not need to train because against any single human opponent, he is more than adequate to the task, but more often, he is fighting against entire armies where one wrong move may be his last.
This is what the training is really for, to prepare him when he is out-manned, out-gunned, or outclassed and prepare him to dig deep to find the hidden reservoirs of awesome he needs to come out on top.
Cap has fought against some of Marvel's heaviest hitters: Spider-Man, Hulk, Thor, The Thing, Mr. Hyde, Wolverine, Magneto, the entire Serpent Society for Pete's sake, the man has fought against a good portion of the Marvel Universe at one time or another.
From his perspective if you're not training, you're dying.
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Some people run just for the joy of running.
My understanding is that it often has to do with endorphins, something that Steve probably doesn't have to deal with in the shape that his body is in, but he might simply enjoy the feeling of the well-oiled machine of his body functioning (especially given he used to lack the sort of musculature or endurance necessary), or he might find it relaxing to think while moving, the same way so many people like to take a walk to think.
And, of course, that's assuming that the super-soldier serum obviates the need for exercise.
The mechanism for it is seldom explained.
The fact that Rogers stepped out of the device fully muscly suggests that it probably has some degree of "instant beefcake", but regular exercise might improve things further, or help with things like muscle definition.
Lastly, he might run simply because it feels like something that normal people do.
The Super-Soldier serum makes him separate from humanity, but that doesn't fit with Rogers's personality.
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Exercising is an essential part of healthy lifestyle.
Super-Soldier Serum gave him muscles, strength and faster metabolism, but it doesn't mean he could stop eating and breathing.
He is still a human who can build up cholesterol in his blood vessels, for example.
There are countless benefits of exercising which I don't think Super-Soldier Serum can provide: Fitness.
Brain Plasticity dictates that it'd support a part of body which is used more often and vice versa.
Your sedentary lifestyle can turn off your motor nerves making you clumsy irrespective of how big your muscles are.
Muscle tissue refreshments.
When you exercise, lots of muscle tissues get broken down and new strong muscles (which can withstand previous stress) are formed using proteins.
New things are always better.
Better blood flow.
Exercising boosts up heart rate to supply more resources (oxygen, nutrition) to involved cells, but the whole body gets benefit.
Also, in the process, a nitric oxide isomer named eNOS is released from the blood vessels into the blood which maintains the blood flow throughout the day.
Fat/Cholesterol burning.
This doesn't need to be explained as heart diseases are popular these days.
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Rogers likely suffers from some form of PTSD, while not necessarily diagnosed with a lifestyle or drug prescriptions, he may be using running as a form of self medication.
Rogers displays common symptoms of PTSD throughout the second film and in the Age of Ultron trailer in self destructive tendencies, distancing himself from others, and distracting himself with other problems, all of which are aggravated by seeing Bucky again and being put under combat-like circumstances.
So overall, Steve probably has PTSD and may use running as a way to make himself feel better as it creates endorphins which help remediate depression, a side effect of PTSD.
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I'm sure he trains because it's a part of the regimented routine the army had him on, and it keeps his life in order.
It's something he can hold on to from his past that hasn't drastically changed since he woke up in the present.
PT is a typical part of the U.S. Military lifestyle which Steve still lives as much as he can.
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If I remember right he loses the power from the super-soldier serum, but I don't remember when it happens.
So he needs to keep exercising to stay in shape.
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In the beginning of Captain America: The Winter Soldier, we see Sam Wilson going for a morning jog and Rogers sprints right by him, "On your left." Then a few minutes later, we see Rogers sprint by Wilson again, and then once more. I liked the whole scene, it was really funny; but why is Captain America exercising? Doesn't his Super-Soldier Serum keep him in tip-top shape anyways?
Why does Captain America need to exercise?
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The Super-Soldier serum did make his body physically fit, but being at the peak of human capacity and being battle-tested are entirely different things.   While we are never told if Captain America needs to exercise to maintain his beef-cake physique, in the comics, the Captain is always training to increase his combat effectiveness and it seems that the more he trains, the more confident he feels about his abilities under fire, especially against entire armies. Given he is one of the members of the team with comparatively limited superhuman abilities, he may feel an extra incentive to push himself to his physical limits. He may also train because he can do things ordinary humans cannot easily do and the only way to reach those peaks of performance is to practice. It may also be a way of dealing with PTSD. While the Super-Soldier serum increases his muscle strength, metabolism and overall fitness, Steve runs to maintain a healthy lifestyle and to push his physical limits and increase his confidence. Moreover, running was part of his regimented routine in the military and he continues to run to keep his life in order.  
The Super-Soldier serum increases his muscle strength, metabolism and overall fitness. He runs to maintain overall fitness, muscle tissue regeneration and a good cholesterol. He runs to combat depression and self-destructive tendencies and to push his physical limits and increase confidence. Running was a part of his regimented routine in the military and he continues to run to keep his life in order. 