It's a bit of a cropped view, but that man isn't really dressed like a monk. However, he does have the traditional markings on his forehead, so it's likely that he is some sort of a devotee.
As for what he's saying: Hare Rama is another chant, parallel to Hare Krishna. Lords Rama and Krishna were two avatars of Lord Vishnu, who was one of the most important gods in Hinduism. Both of their names are parallely invoked by devotees, generally to mark an occasion or to ask for help. It's not really a salutation, which is a greeting- It's used in somewhat similar situations that Christians may use the sign of the cross for. The oldest recorded version is from a text known as the Kali-Saṇṭāraṇa Upaniṣad, and it is as follows:
Hare Rama Hare Rama
Rama Rama Hare Hare
Hare Krishna Hare Krishna
Krishna Krishna Hare Hare
(There is some debate as to what the Hare refers to; some think that it is a version of another name for Vishnu, Hari, while others think that it is a version of Hara, which is another name for Krishna's consort Radha [Source])
NPR analyses the spiritual significance:
The word "Hare" refers to the divine feminine potency of God. "Krishna" means the all-attractive one, and "Rama" is the reservoir of all pleasure.
Hare Krishnas believe that the sound vibration of the mantra has a direct impact on the soul. According to a philosophy of ancient India, the soul is spiritually asleep. Just as an alarm clock awakes a sleeping person, the Hare Krishna mantra awakens the soul to its spiritual reality — whereby it can experience its eternal connection with Krishna or God. And devotees believe that a person need not understand the language of the mantra, because the sound vibration transcends the sensual, mental and intellectual levels of consciousness and puts one directly in touch with the spiritual.
You can listen to a repeated version of the mantra here on Youtube.
Read the full Wikipedia article here. It's mainly on Hare Krishna, but it has some information about the other part too. See this, too.
Why is the guy in the film saying it? I'm not sure. I took a screenshot of the flyer, and it doesn't have Rama's name on it- only Krishna's. So it's a bit strange that he would say what he did. The guy also pronounces it with an American accent and sounds pretty sarcastic. That's definitely not a normal usage of Hare Rama; I suspect he was saying it in the same manner that you may sarcastically tell someone to Have a nice day, or I love you, too after they've been rude to you. The woman's response supports that, but that's definitely not a typical usage.
(Also, monge is Portuguese, isn't it? From what I know, that man isn't a monge, as they generally live in secluded monasteries. That man's more of a Swami or a Bhaktar)