Learn Russian: У Ивана и Анны есть хороший дом.

Because they are the owners, they must be in the Genitive case after the word У. For masculine names ending in a consonant like Иван, we add -а to get Ивана. For feminine names ending in -а like Анна, we change the -а to -ы to get Анны.

How exactly does У ... есть mean 'to have'?

Russian rarely uses a direct verb for 'to have'. Instead, it uses a structure that literally means 'By [someone] there is [something]'. The word У means 'by', and есть means 'is' or 'exists'. So this sentence literally translates to 'By Ivan and Anna there is a good house'.

Does the word есть change because there are two owners?

No, есть stays exactly the same. In this sentence structure, the owners (Ivan and Anna) are not the grammatical subject doing an action. The word есть just means 'exists', and it does not change regardless of how many people own the item.

Why don't the words хороший дом change their endings too?

In Russian possession, the thing being owned is actually the subject of the sentence! Since the sentence translates literally to 'By Ivan and Anna there is a good house', the phrase хороший дом (good house) is the subject and stays in the Nominative case, which is its basic dictionary form.