Learn Russian: Применяемый метод объясняет это явление. - The applied method explains this phenomenon.

It comes from the imperfective verb применять (to apply). To form the present passive participle, you take the first-person plural present tense form (мы применяем, "we apply") and add a standard adjective ending. Since it describes the masculine nominative noun метод, it takes the ending -ый.

Could I use the active participle применяющий instead?

No, because that would mean the method itself is actively applying something else! Because the method is the thing being applied by someone, we must use the passive participle применяемый.

Why is применяемый translated as "applied" if it is a present participle?

In Russian, present passive participles literally mean "being [done]". So применяемый метод strictly translates to "the method being applied" right now. In natural English, however, we often just say "the applied method" to express this.

Why doesn't the ending of это явление change, since it is the direct object?

It actually is in the accusative case, as required by the verb объясняет (explains). However, явление is a neuter noun, and for all neuter nouns (and their modifiers like это), the accusative case looks exactly identical to the nominative case.