Learn Dutch: Eenzamen hebben vaak een huisdier. - Lonely people often have a pet.

This is a nominalized adjective, which means an adjective being used as a noun. When an adjective is turned into a plural noun to refer to people (meaning "lonely people" here), it gets the -en ending. You could also say eenzame mensen, but eenzamen is a more concise and very common alternative.

Why did the spelling change from a double 'aa' in eenzaam to a single 'a' in Eenzamen?

This is a great time to refresh a core Dutch spelling rule! In the base word eenzaam, the 'aa' is in a closed syllable (it ends with the consonant 'm'). When you add the -en ending, the syllable breaks into an open syllable (een-za-men). A single vowel at the end of a syllable naturally makes a long sound, so we drop the extra 'a' to avoid double-spelling it.

Shouldn't there be an article like De before Eenzamen?

Not in this context. Just like in English ("Lonely people..."), Dutch leaves out the article when making a general statement about an entire group. If you said De eenzamen, it would translate to "The lonely people" and imply you were talking about a specific group of lonely people you had already mentioned.

What is the literal translation of huisdier?

It is a compound noun made up of huis (house) and dier (animal). So literally, it means "house animal", which is the standard Dutch way to say "pet".