To straightening curly hair weave you will need either a straightening comb and/or a flat iron. First, of course, wash the weave hair and condition. Blow dry the hair weave as straight as possible without disturbing the weft. If you have a dome hair dryer, dry your roots by sitting under that.
Once your hair is completely dry (make certain it is completely dry, because if you don't your hair will appear "puffy"), apply a small amount of a very light hair maintenance product. Then begin using your straightening comb and/or flat iron to straighten the curly hair weave.