Learn Turkish: Masadaki silgi ve zımba yeni. - The eraser and stapler on the desk are new.

The current lesson covers using -(y)AcAk to describe what an object is used for. Why don't we use that grammar here to say 'things to erase and staple with'?

Because silgi (eraser) and zımba (stapler) are already dedicated, everyday nouns. You would use the -(y)AcAk structure, like silecek bir şey (something to wipe/erase with) or zımbalayacak bir alet (a tool to staple with), when you need to describe an object's utility because you forgot its name or are improvising a tool.

What exactly does the -ki in masadaki do, and why can't we just use masada?

The suffix -ki attaches to the locative case (masada) to turn 'on the table' into an adjective that directly modifies the nouns after it. It creates the meaning 'the on-the-table eraser and stapler.' If you just said masada silgi ve zımba yeni, it would sound grammatically disconnected, somewhat like 'On the table, eraser and stapler is new.'

Since 'eraser and stapler' makes a plural subject, why isn't the adjective yeni pluralized as yeniler?

In Turkish, inanimate plural subjects (things, concepts, animals) generally take a singular verb or predicate. You only pluralize the predicate (like adding -ler to make it yeniler) if the plural subjects are humans or if you are deliberately personifying the objects.

Where is the verb in this sentence?