I feel like I'm reasonably good at picking at a game on the gameplay level, as per what works and does not and why and surface videogame essayist stuff like ludonarrative dissonance (or the rare examples of ludonarrative harmony).
I may offer you my finest insight into video games such as "Lara Croft has some sort of father complex going on" and "Shadow of Chernobyl is unintentionally about life in the collapse of the soviet union" which even by my own admission feels shallow and trite. You watch someone like Jacob Geller or Noah Caldwell-Gervais and they have fascinating things to say even on games you wouldn't expect it, like NCG on Quake.
How do I become that knowledgeable? Interesting? Analytical? about video games?
Since others have already offered a lot of good advice, I'll go in a bit of a different direction.
Game stories are often quite shallow, with fairly surface level themes, because they need to proritise the "game" part and often need to alter the "story" part in order for the game to still play well. Understanding game development and how it works can help a lot with understanding why a video game might be the way it is. Understanding the way a certain medium is used to create art can help you understand why that art is the way it is, due to the limitations and benefits of the medium.
I'm mostly trying to get better to weave connections between the gameplay and the story. Like for story reasons alone I wouldn't bother much analyzing Tomb Raider on account of there isn't a lot there. It's an action movie plot to get you to go to next location and try to make some characters sympathetic. More like how the survival elements are represented in both story and gameplay, allthough entirely superflous in the latter past the tutorial. Not even in "optional side content that gives you cool stuff" until you hit Rise.