Learn Russian: Дети любят кошек и собак.

They are the direct objects receiving the action of любят (love), so they need the Accusative case. Because they are plural animate nouns (living things), their Accusative form is exactly the same as their Genitive plural form. Both words drop their final -а.

If кошка drops its final -а, where does the -е- in кошек come from?

When кошка drops its ending, it leaves the awkward consonant combination "шк". Russian inserts a "fleeting" vowel -е- to break up the consonants and make the word easier to say.

Why doesn't собак get an extra vowel inserted like кошек does?

When собака drops its final -а, the word ends in "собак". Since "бак" is already a normal, easy-to-pronounce syllable, no extra fleeting vowel is needed.

Why doesn't the word Дети (children) take this animate Accusative ending too?

Дети is the subject performing the action (they are the ones doing the loving), so it stays in the Nominative case. The Accusative case is only applied to the objects being loved.