It's a Late Medieval tournament helmet. For sports, rather than for war.

The early forms of the tournament were very war-like, often fought with the same equipment as used for war: the same armour, the same sharp swords. Some tournaments were not just training for war, but actual war: pre-arranged battles fought according to agreed rules at an agreed time and place, often with spectators (sometimes, this kind of battle was simply a normal part of the local tradition of warfare, and were just considered normal battles rather than tournaments, despite including sports-like features such as the erection of grandstands for the spectators, and team size limits). The more extreme form of such tournaments could even include rules banning the use of armour.

Over time, the tournament changed into a safer and more controlled sport, with the use of blunted weapons, and more restrictive rules, and became a warlike-sport, providing a relatively safe opportunity for the display of martial prowess and pageantry ("relatively safe" in comparison to warfare and earlier tournaments - it was not risk free, as seen by Henry VIII's (King of England) injury in 1536 in a fall from his horse during the joust, and Henri II's (King of France) death in 1559 from lance splinters through the eyeslot of his helm). Part of this change was the development of specialised tournament armour and weapons. Armour included tournament armours that were more protective, at the cost of being to heavy for practical use in battle. Tournament armours included both armours completely purpose-built for the tournament, and reinforcing pieces that could be worn over normal battle armour.

The helmet you show is made specifically for tournament use. The grilled/barred visor gives excellent visibility and ventilation, and is sufficient protection for the melee (and would be sufficient against sharp weapons, apart from thrusts by slender daggers and slender-pointed swords (and of course arrows and crossbow bolts)). This kind of helmet is called a Kolbenturnierhelm (German for "club-game helmet"), or Kolbenhelm, or Turnierhelm. It's a variety of great bascinet, with the "great" referring to the types with the gorget (the neck protection) permanently (and rigidly, by some definitions) attached to the rest of the helmet. This restricts the mobility of the helmet, and is much more suited for tournament than war.

An example of a great bascinet that has a pierced rather than barred/grilled visor: https://commons.wikimedia.org/wiki/File:Visored_Bascinet_MET_04.3.240_008june2015.jpg

Two examples of helmets similar to the one you show:

https://www.metmuseum.org/art/collection/search/34276

https://www.metmuseum.org/art/collection/search/24678

These helmets are a Late Medieval development. They prominently feature in the equipment shown in the mid-15th century Le Livre des tournois (Book of the Tournament) by René of Anjou.

From this book, a similar helmet, complete with fanciful crest: https://commons.wikimedia.org/wiki/File:Traiciti%C3%A9_de_la_forme_et_devis_comme_on_fait_les_tournoys_BNF_Fr._2695_f23v-heaume3.jpg

The weapons shown in your linked picture are also specialised tournament weapons. The sword, held in the fighter's hand, is blunt and broad-tipped (reducing the risk from (accidental?) thrusts), and is possibly made of baleen (whalebone, but not the bone kind of whalebone). The club, hung from the saddle, is a safer substitute for a mace. Both weapons are shown in René's book: https://commons.wikimedia.org/wiki/File:Traiciti%C3%A9_de_la_forme_et_devis_comme_on_fait_les_tournoys_BNF_Fr._2695_f31v-armes.jpg

For such helmets and weapons in use of the playing field: https://commons.wikimedia.org/wiki/File:Turnierbuch_des_Ren%C3%A9_von_Anjou_23.jpg (note the club-wielder near the bottom left, and the fighter with a human-faced crest like the one in your picture just below the bottom left of the pink spectator stand).

Links to a digitised version of René's book can be found at https://en.wikipedia.org/wiki/Le_Livre_des_tournois (and much of the art is available via https://commons.wikimedia.org/wiki/Category:Traicte_de_la_Forme_de_Devis_d%27un_Tournoi including the ones linked above).

These barred/grilled helmets are unsafe for jousting, due to the risk of splinters from broken lances. The classic Late Medieval jousting helmet was the frog-mouth helm:

https://www.metmuseum.org/art/collection/search/23216

http://britishlibrary.typepad.co.uk/.a/6a00d8341c464853ef01901de3af20970b-500wi - from http://blogs.bl.uk/digitisedmanuscripts/illuminated-manuscripts/page/56/

The good protection this provides can be seen in the main frontal photo of this helm: https://www.metmuseum.org/art/collection/search/27128

Further examples of sporting (and battle) armour can be found in Matthias Goll's PhD thesis, Iron Documents: http://www.ub.uni-heidelberg.de/archiv/17203 DOI: 10.11588/heidok.00017203

Barred helmets continued in use in post-Medieval times, notably in the Italian Gioco del Ponte (game of the bridge), a tournament-like battle fought on a bridge between two teams. The helmets used were often old military helmets with visors replaced by bars: https://www.metmuseum.org/art/collection/search/27092 (see the references given on the Met's page for that helmet for more on Gioco del Ponte). And, of course, we still see barred/grilled helmets in modern sports, from protection from balls and sticks of various kinds.