Please tell me whether the verbs in the following two sentences are transitive or intransitive.

1. I looked down from the window.

2. The dog ran after me.

Please also give the reason behind the answer.

  Yes, both of your sentences are transitive as the first one answers the question WHAT  and the second one answers Whom .

Both the sentences are transitive.....

Ofcourse, it will be transitive beacause, the noun or pronoun that receives the action, and answers the questions ‘what?’ or ‘whom?’). In this case, the action is passed on from the doer or subject to the receiver of the action or the direct object.