Also from Wikipedia:
The copyright nature of "Let's Play" videos remains in question; while the developer and/or publisher of games typically possess the copyright and granted exclusive distribution rights on the media assets of the game, others cite fair use claims for these works as their nature is to provide commentary on the video game. In one case, Nintendo claimed that they retain the copyright and have registered the content through YouTube's Content ID system such that they can generate ad revenue from user videos, though Nintendo would later back off of such claims, and later created its own affiliate program between themselves, Google, and proactive uploaders to split profits. Smaller developers have been more open to allowing Let's Play videos. Ubisoft has stated that it allows its games to be used in Let's Play videos and allows for those making them to monetize from any ad revenue as long they stay within certain content-appropriateness guidelines.
In early December 2013, a change in YouTube's ContentID policy caused many existing Let's Play and other video-game related material to be blocked. In response, many developers and publishers issued statements and worked with YouTube to assure such videos were not meant to be blocked, helping those whose videos were affected, and encouraging users to continue to show these; these companies included Blizzard, Ubisoft, Capcom, Paradox Interactive, and Valve. YouTube later clarified that the change in the ContentID system that caused videos to be flagged was likely a result of new tools it made available for multi-channel networks, which can cover separate video and audio copyrights. At least two known music multi-channel networks, TuneCore and INDmusic, who represent many video game music composers and artists, had automatically enabled the copyright protection for all of its clients without seeking their input, and as such, many of the Let's Play videos as well as the game developers' own promotional videos were blocked due to these actions. YouTube states they do not plan to change this system despite complaints from the original music composers
I've seen many YouTubers cite Fair Use for commentary to justify what they are doing. Usually is it listed in the description of the video, "images used under Fair Use" or something like that.