I have a few hypothesis on the matter, but I couldn't find any details on how and why those equipment came about.
I've seen artistic depictions of them sure, and they seem to have been used by a good number of these mercenaries in combat (although I have seen bare-headed combatants in those drawings as well, so I remain doubious of the accuracy of those works). But did they really wear them to battle? I mean there is still a very real risk of getting a pointed stick to their faces, wouldn't a helmet with cheek guards been better?
A reply to /u/lcmiracle
The Landsknechte were a unique group, not only because of their means for conducting war -- combining firearms with pikes -- but also because of their social status. The twilight of the 1400s saw the rise of infantry at the expense of the armored (mounted) knights' supremacy in military thinking in Europe. I've written here and here on how the Swiss pikes challenged French gendarmes, and here on how the nature of armies changed in that time period.
Unlike the knights, the Landsknechte were recruited not exclusively from lower nobility; they were initially mostly peasants from mountainous regions where there were a lot of population pressure, or from regions that had had nature or man-made catastrophe. As such, they refused to bow to the social order of knights. To quote Olaf van Nimwegen's The Transformation of army organization n early-modern western Europe:
Becoming a Landsknecht meant leaving normal society and joining an organisation that, similarly to a monastic order, lived by its own rules. This does not of course mean that soldiers and monks lived a comparable life; on the contrary. Monks were supposed to be pious and chaste, whereas Landsknechte emphasised their manhood as expressed by their unruly behaviour and extravagant clothing.
The Landsknechte were known as free soldiers in this sense. They were not bound to the structures of the knight. Neither did they think of themselves as bound to other social structures of the time. So they were free soldiers who worked for the highest bidder regardless of political affiliation. This is not so say they were unreliable, rather to the contrary they were very disciplined in battle. They elected their own officers and individually gave oath to military justice. They had a strong group identity and fidelity. In a way, they transformed the guild system into one for the military, and one that wasn't bound to a city or locale either.
The flamboyant clothing and hats were a display of this identity. These articles of clothing served no real protection in battle. That's why in reality, they differentiated what they wore depending on the needs. For example, armored pikemen wear (partial) armor and appropriate helmets, even if many still wear the colorful sleeves and pantaloons.
The Landskechte were a unique military system that illustrate the radical changes happening in Europe during the early modern era!