This is a Japanese PlayStation 4 launch exclusive item (Feb 22, 2014 with the Japanese release of the PS4) available as a free download from the Japanese PlayStation Store. This page provides info on how to create a Japanese PlayStation Network account.
Note that each Sony Entertainment Network Account can be activated as your primary PS4 system. See this support article for additional information and an explanation of how activation works on the PS4 (it is different from how that worked on the PS3.) The simplest explanation is if this is your only PS4 system and you activate each local account that you create on your PS4 as the primary, any game, app, and content downloaded on any other account, will be usable by other local accounts on that PS4.
In the case of HGOS, if you activate the PS4 as the primary for your Japanese PSN account, any other local account on your PS4 will be able to run HGOS (and any other content downloaded on your Japanese PSN account.) This does not mean that your Japanese PSN is now the primary account for your PS4 (again, see the above support article for an explanation of what primary system means when it comes to the PS4.)
Host Girls on Stage (HGOS) is what is usually being streamed on services such as Twitch, Ustream, or NicoNicoLive (this shows up as a viewing and streaming option when logged in to a PS4 account that is associated with a Japanese PSN account.)
The free game comes with one song (恋・KOI☆week end!) and a set of outfits. The game also provides the ability to easily earn 14 Bronze trophies (by rotating each girl as the lead singer) and 1 Gold trophy (this secret one is awarded at the end for having all 14 girls sing lead.) See the game play guide for more information regarding navigation as well as a link to a trophy walkthrough.
The first time you run the game, it will prompt you to create a game save file (you'll notice the following katakana in the huge wall of text セーブ). Select Yes. You will then be presented with a name entry screen (L3 toggles between kana and English romaji input). The default name is "ドリムチ" but it does not matter what character name you use here since it isn't used (unlike in the host club sims.) Always use "O" for your selection/character entry since "X" will cancel/back you out (the game uses the Japanese input convention even in the text entry part). Maximum amount of characters (kana and English) in a name is around 6. R2 ends text entry (you may need to do R2 twice) and the game will prompt you if this is correct (select "はい" and press "O" to create your game save.)
The majority of its content is in the form of DLC's (27 additional songs and numerous outfits). Much of this content is a reuse of songs that could be unlocked in the previous games and/or purchased as DLC's (HGOS in general reuses a significant amount of content from the actual games except for hair arrangements and accessories.) In order to reposition each character or to set them in specific groupings for team singing, this also requires a "designation rights" DLC (which can be purchased individually per character at ¥100 each or purchased at a discount unlock of all hostesses for ¥1000.) This also unlocks the ability to set each hostesses "expression" (sober or tipsy). It's best to just get the ¥1000 set.
Content on the Japanese PlayStation Store can be purchased by filling your account with PlayStation Network point cards (yen). Credit cards can only be used if it is a Japan issued one (which means you need to have a Japanese billing address.)
The actual game play in this is very minimal; outfit changes, the before-mentioned changing of singer positions, lead singer, solo or team singing, and ability to control the camera manually (or to set it in a free-roam mode or stationary position.) Songs can be set to play in order or randomly shuffled. The options menu also allows for setting the vocal levels of each individual hostess (there is a default options setting as well one per song; the latter being adjustable when hitting the Option button while a song is playing, and going into its "Option menu" to set the levels.
Because there is very little game play, most folks have only a very tiny view of each host girls personalities (if the majority of their exposure to the DCC franchise is from HGOS.) Once this site is built out more, hopefully some of that information will be useful in lieu of playing the games (which will be difficult for many due to the language barrier.)
The karaoke aspect of Host Girls on Stage comes from the other Dream C Club games where one of the mini-games is karaoke (both during story mode and as part of the VIP room unlock which allows for a karaoke mode that is similar to the one in HGOS but with no more than 2 host girls at a time). Each hostess has their own "character" songs where they are the primary singer (these songs also contain hints and information about that specific character which can be helpful in the context of the Dream C Club games.)