What kind of a grammatical word is 'have'?

In the context you have used, 'have' is, like 'had', 'were' 'are', 'is, 'will', and more, a part verb which combines with the participle 'broken', to make a whole verb in the present tense. So your sentence is using 'have' to support 'broken' - which cannot be used alone - to form the present tense. If you were talking about something that had happened in the past, you could say, "The pictures 'were breaking' my heart, until I discovered that they were fakes". For the future tense, you could write, "If ever I see the pictures, surely they 'will break' my heart"; and so on.