As someone who grew up on 24th-century Trek -- and, in general, on more recent television -- I find it helpful to think of TOS almost as theater.
Star Trek in general is very theatrical, but TOS especially so. For instance, start with the dialogue. With the exception of Enterprise, none of the Star Trek series really tried to make their dialogue sound realistic to contemporary ears. One advantage of this is that it keeps the dialogue from becoming painfully dated; there's no "groovy" slang or anything other than correct, formal English, something that changes very slowly. However, it also gives the dialogue a theatrical, oratorical feel. (It helps that William Shatner, Patrick Stewart and Avery Brooks all did extensive Shakespere work before Trek.)
Thinking of it as theater also helps when you're watching a '60s show as someone who's become used to modern TV techniques. In a stage play, nobody really expects hyper-realism in special effects. Many things are suggested rather than shown, and the convention is that you simply suspend disbelief -- ignoring, for example, the proscenium, the visible lighting positions, the actors playing multiple unrelated roles, or any scene changes that may happen with the curtain open. Something like the Gorn or the Horta looks ridiculously fake when you view it through the same lens that you'd view modern TV, but you wouldn't think twice about it on stage.
TOS in particular has a very stylized feel -- everything from the bright colors of the bridge to the way characters are lit to the music cues, which are much more obvious than in any Trek that was made after Season 2 of TNG. Don't expect it to be realistic; think of the lighting and sound as storytelling conventions.
If you've ever seen a show like "West Side Story" or "Les Miserables," try thinking of it more like those. Musical theater in particularly is a very stylized genre. Nobody believes that gangs in New York in the '50s went around spontaneously breaking into elaborately choreographed dance routines, or that heroes and villains in 18th-century France went around singing to each other. But those are the conventions the genre uses to tell its stories, and both shows, when done right, are very powerful pieces of theater.