Military traditions arise naturally in a society dedicated to war. They permeate its culture and create an atmosphere of reverence towards all soldiers and their lives. As a result, those that become part of the military feel like combat experience is not really new to them, that they half-know already all that could happen to them on the battlefield. And this helps them achieve veterancy at an incredible rate.
Military tradition is an incredibly useful Policy, especially for conquest-oriented empires! Adopt it as soon as you intend to embark on your first conquest to see your units piling up promotions like crazy! Thanks to this Policy, it is fully possible to have an army possessing the top-level abilities as early as the Renaissance Era, and needless to say such an army is terrifyingly more effective than one composed of inexperienced regular units.
That being said, you shouldn't waste a Policy slot before you're ready for that conquest. Fighting Barbarians yields a maximum of two promotions (if you don't use one level-up to Heal!), and the only possibility to continue acquiring Experience is to fight civilizations and/or City-States.
Military tradition is the policy of honoring and romanticizing service in the nation's military. Soldiers and veterans are venerated, while those who do not serve in the military are inferior, possibly even dishonorable. Modern Israel has a significant military tradition, as did Victorian England and the pre-Civil War southern United States.