Learn Finnish: Jos sinulla on kiikarit, näet piiloutuvan ketun. - If you have binoculars, you see the hiding fox.

It comes from the verb piiloutua (to hide oneself). You take the strong verb stem (piiloutu-) and add the active present participle ending -va. Since it acts as an adjective describing the fox, it must match its case and takes the object ending -n.

Could I say näet ketun, joka piiloutuu instead?

Yes! Using a relative clause with joka (who/which) means exactly the same thing. The active present participle is just a very common, compact way to turn an action into an adjective describing what someone or something is currently doing.

Why do both piiloutuvan and ketun end in -n?

The fox is the complete object of the action näet (you see), so kettu takes the -n object ending (often called the genitive-accusative). Because piiloutuva is acting as an adjective attached to the fox, it has to agree with the noun's case and also takes the -n ending.

Why is kiikarit in the plural nominative form instead of the partitive?

Binoculars are treated as a plural word in Finnish (kiikarit), just like glasses or scissors. In the "someone has" structure (sinulla on), the thing you have is usually put in the nominative case. For plural words, this means taking the -t ending.