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?
Ask questions (to yourself, but to other people works too). Why did the author make this decision? Why are things this way? If you ask "why" questions you'll come up with your own interpretation of the game.
You just kinda have to spend a lot of time thinking about it. There are some pieces of media I've really liked for a long time and as a result I have fairly developed views on them just from the length of time I've been thinking about them.