Learn Russian: Благодаря разработчикам программа работает быстро и без ошибок. - Thanks to the developers, the program works quickly and without mistakes.

Because it follows the preposition благодаря (thanks to). This preposition always requires the Dative case to show who or what is responsible for the positive outcome. Here, the plural noun разработчики (developers) changes to its Dative plural form, разработчикам.

Could I use из-за (because of) instead of благодаря in this sentence?

You shouldn't. While both express a cause, благодаря is used specifically for positive results. Из-за is typically used for negative outcomes or placing blame. If you used из-за разработчикам, it would sound like you are sarcastically blaming the developers for the program working so well!

Благодаря looks a lot like the verb благодарить (to thank). Are they related?

Yes, exactly! Благодаря was originally a verbal adverb (gerund) formed from благодарить, literally meaning "while thanking." Over time, it evolved into a standard preposition, which is why it still carries that positive, grateful meaning.

Why does ошибок look so different from its dictionary form ошибка (mistake)?