Learn Icelandic: Ég man ekki nýju hlaðvörpin. - I do not remember the new podcasts.

It is the weak plural form of the adjective nýr (new). When we talk about specific things, like "the podcasts" with the attached definite article, the adjective becomes weak. The great news for the plural weak declension is that the adjective always ends in -u regardless of the noun's gender or case!

Is the verb man (remember) causing the words after it to be in a specific case?

Yes, the verb muna (to remember) forces its object into the accusative case. Therefore, nýju hlaðvörpin is in the accusative plural. For neuter words like hlaðvarp, the nominative and accusative forms are identical, which makes this one easy to handle.

Why is there an ö in hlaðvörpin if the singular word for podcast is hlaðvarp?

This is due to a vowel shift you might remember from earlier lessons called U-mutation. When the neuter noun hlaðvarp is made plural, the a shifts to ö, making the plural form hlaðvörp. Then we add the neuter plural definite article -in to make it hlaðvörpin (the podcasts).

Why is there no separate word for "the" before nýju?

In everyday Icelandic, you express "the [adjective] [noun]" by using the weak form of the adjective followed by the noun with the definite article attached to its end, just like nýju hlaðvörpin. Adding a separate free-standing word for "the" before the adjective exists, but it sounds very formal and is mostly used in written text.