It is not complex and doesn't require much background knowledge ("critically" reading a couple books like So You Want To Talk About Race is enough, or you can just skip those and read Cynical Theories). This "critical sense of the term" is basically a political wedge. There are two almost completely unrelated issues at play:
1) Institutions, like in most of the world, sort people by class and reward success; therefore, they may propagate class divisions and allow (or even amplify) differences in achievement.
2) Due to the history of the USA, the class in the USA is tied to race to a considerable degree.
(2) can and should be solved separately with policies ranging from affirmative action to reparations/better redistribution; however, this is not what (or not everything that) activists want. Books like New Jim Crow for example are explicitly ambivalent about affirmative action, because it allows some people success while leaving the rest of the poor behind and distracting from the struggle/compassion for them.
Leftists instead bundle those together and use it as a wedge against unsuspecting/apolitical center - if you are against "solving" (1) with leftist policies, you must be a-ok with (2) and therefore you are racist. When in reality, (1) in isolation is one of the greatest things for a society and should not be "solved" at all. Kendi is the most explicit and obvious example of this that comes to mind, others may be less explicit about it.
It's important for Americans to distinguish race from class, and to understand that, just like in the rest of the world, inequality as such is totally fine, and in fact good for society; "equity" is evil beyond school age, as proven thru history every time it has been tried. I grew up ~poor in Russia surrounded by other ~poor people, all white; that made me extremely classist. I personally cannot wait for a wonderful day where every kid has a chance to make a life for themselves, and a representative (M% white, N% black, etc.) upper-middle class in America can look with a slight pity and disdain at similarly (M% white, N% black, etc.) poor and say "you've had your chances, so why cannot you be more like me?" :)