> Why don’t the textbooks explain why the algorithm is correct?

The good ones do!

> Should I be familiar with every step of Dijkstra’s search algorithm and remember the pseudocode at all times?

If it’s the kind of thing you care to be familiar with, then being able to rederive every step of the usual-suspect algorithms is well within reach, yes. You don’t need to remember things in terms of pseudocode as such, more just broad concepts.