tempesta has a metaphorical meaning for: " big trouble in my heart " or " problems in life" He or she (I think he) is talking about his love, but the feeling itself is described as 'pain' (dolore), so 'dolore' should be seen as an adjective to what follows. Actually there is a syntax problem: a possible fix is this: " è questo dolore, è questo amore che ho per te, che mi fa superare queste vere tempeste " Of course when there is a problem in the syntax could be many fixes, with slightly different meanings. At first sight it looked to me as a lyrics excerpt, but actually I don't think it is , or else it can't be faithful to the song; anyway Italian singers and authors can be very very good, and others can be quite bad and sloppy, regarding to lyrics. On the other hand if the writer is learning Italian this is definitely a good try!

"It is for this pain, for this love I feel for you, that what makes me overcome those real storms " ( literally storms, but metaphorically and figurative for turmoils, trouble and pain of being in love)