Learn Russian: Я знаю этих популярных блогеров.

Because блогер is an animate noun (a person). In the plural, the Accusative case for animate nouns and their matching adjectives is exactly the same as the Genitive plural. Since the bloggers are the direct object of the verb знать, they are in the Accusative case, but they borrow the Genitive endings -их, -ых, and -ов.

How would this sentence change if I wanted to say 'I know these popular blogs' (inanimate)?

If the noun is inanimate, the Accusative plural matches the Nominative plural instead of the Genitive. You would say Я знаю эти популярные блоги. Notice how the words stay in their basic plural forms because a blog is not a living being.

What is the Nominative plural form of this exact phrase?

If these people were the subject of the sentence (for example, 'these popular bloggers know me'), you would use the Nominative plural form, which is эти популярные блогеры.

Why does этих take the ending -их but популярных takes the ending -ых?

This is just about hard and soft spellings. The special word этот has soft endings in the plural, giving us этих. Regular adjectives like популярный have a hard stem (ending in the hard consonant н), so they take the hard ending -ых.