Learn Finnish: Pieni koira oli karannut puistoon. - The small dog had run away to the park.

The pluperfect (had done) is built just like the perfect tense (has done), but you use the past tense of the helper verb olla. So instead of on karannut (has run away), you use oli karannut (had run away).

When would I use this pluperfect form instead of the simple past karkasi (ran away)?

You use the pluperfect to show that an action was already completed before another event in the past. Saying oli karannut implies a context like, "By the time we opened the gate, the dog had (already) run away to the park."

How do we get the form karannut from the basic verb karata (to run away)?

Karata is a Type 4 verb (words ending in -ata or -ätä). To make the past participle (the -nut form) for Type 4 verbs, you generally add -nnut to the stem. The double nn happens because the t from the verb's core stem assimilates to match the n in the ending.

Why is the dog running puistoon (into the park) rather than using a case like puistossa (in the park)?

In Finnish, verbs of movement focus strongly on the destination. With the verb karata (to escape, to run away), you use the illative case (the "into" case, ending in a doubled vowel + n) to express where the subject ended up after escaping.