This pattern is commonly used when identifying or defining something. It's almost like saying "A spider, it/this [is] an animal".
Some more examples from Modern Hebrew: An Essential Grammar:
מי זה יורם?
Who [is] this, Yoram? ("Who is Yoram?")
יורם זה השכן
Yoram, this [is] the-neighbour ("Yoram is the neighbour")
מה זה ג׳יפ?
What [is] this, a-jeep? ("What is a jeep?")
ג׳יפ זה אוטו
A-jeep, this [is] a-car ("A jeep is a car")
(Apologies if the question marks aren't in the right place. I have a feeling this is gonna happen here a lot!)
The "is" omitted in Hebrew, as you've seen, is before an adjective.
The door is green - הדלת ירוקה
When describing generalities (a spider, an insect), Hebrew would have זה/היא/הוא, like this sentence.
I know חיות is the plural, but what is the difference between חיה and חייה? That's the word used in 'pet' חייה מחמד. Thanks in advance.
A pet is a חית מחמד. Could be you saw it as חיית מחמד, it is used sometimes (probably to show the י isn't a vowel). The word חייה would be pronounced "kha-yeha" and means "her life"