Learn Russian: Кот прячется под ковром.

The preposition под (under) takes the instrumental case when describing a fixed location, answering the question "where?" (где?). Because the cat is already located under the rug, we use the instrumental. If the cat were actively running to a spot under the rug (answering "to where?"), we would use the accusative case instead.

Where did the -ё- in ковёр go when it became ковром?

Ковёр has what is called a "fleeting vowel." When adding case endings to certain masculine words (like ковёр or отец), the last vowel in the stem drops out. So, the stem ковёр- shrinks to ковр-, and then we add the standard instrumental ending -ом.

Why does the verb прячется end in -ся?

The base verb is прятать, which means "to hide [something else]". Adding the reflexive suffix -ся makes the action reflect back onto the subject. So кот прячется literally means "the cat hides itself."

How should I pronounce the -тся ending in прячется?