Learn Italian: Quella superstizione è irrazionale; non dovresti più crederci. - That superstition is irrational; you shouldn't believe in it anymore.

It translates to "in it". In Italian, you "believe to" something (credere a qualcosa). The pronoun ci is used here to replace a quella superstizione (to/in that superstition) so you don't have to repeat it.

Why is ci attached to the end of the word instead of coming before the verb?

Because it is attached to an infinitive (the dictionary form of a verb). To attach a pronoun to an infinitive, you drop the final -e and stick the pronoun on the end (credere + ci = crederci). Because there is a helping verb (dovresti), you could also put the pronoun at the very beginning: non ci dovresti più credere.

Why use dovresti here instead of devi?

dovresti is the conditional form, meaning "you shouldn't" or "you ought not to". It is used to give advice. If you used the regular present tense non devi, it would be a strict command meaning "you must not".

How do you say "not anymore" in Italian?

You wrap the verb with non and più. The non goes before the verb, and più goes directly after it. Here, non dovresti più means "you shouldn't anymore".