Learn Russian: Прибывший охранник увидел сбежавшего преступника. - The guard who arrived saw the escaped criminal.

They are active past participles formed from the perfective verbs прибыть (to arrive) and сбежать (to escape). Because both infinitives end in a vowel plus -ть, you simply drop the -ть and add the suffix -вший (which then changes its final letters to match the noun's case).

Why does сбежавшего end in -его while прибывший ends in -ий?

Participles decline exactly like adjectives to match the noun they modify. Прибывший modifies the subject охранник, so it takes the nominative ending -ий. Сбежавшего modifies the direct object преступника. Because "criminal" is an animate masculine noun, its accusative form matches the genitive, so the participle takes the animate accusative ending -его.

Why do these participles use perfective verbs instead of imperfective ones?

Active past participles made from perfective verbs describe a completed action that happened before the main action of the sentence. This perfectly captures that the guard had already arrived (прибывший) and the criminal had already escaped (сбежавшего) prior to the exact moment the guard saw him (увидел).

How should the ending of сбежавшего be pronounced?