The Tur (Rabbenu Yaakob Ben Asher, Germany-Spain, 1269-1343) writes (Orah Haim 281) that after reciting "Az Yashir" on Shabbat morning, it is customary to recite the special "Nishmat" prayer which gives praise to G-d. The Lebush (Rav Mordechai Yaffe, 1530-1612), citing from the Zohar (Vayakhel, 205b; listen to audio recording for precise citation), explains that this recitation was instituted because of the "Neshama Yetera" – the "extra soul" with which we are endowed on Shabbat. As we receive an extra soul on Shabbat, we recite this special prayer to give praise to G-d.
Noting the special significance of this prayer, the Hida (Rav Haim Yosef David Azulai, 1724-1806), in his work Tov Ayin (18:35), writes that as this prayer is mentioned in the Zohar, it must have preceded even the Mishnayot, and is thus a very ancient prayer. The Hida adds that the text of "Nishmat" is very deep and contains numerous references to the sacred Names of G-d.
Rav David Pardo (1718-1790), in his Michtab Le’David (Orah Haim 13), writes that the recitation of "Nishmat" serves to prepare us for receiving the extra soul of Shabbat.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that there is a "tried and tested" tradition that a person experiencing some kind of crisis should commit to assemble ten people and recite "Nishmat" to praise G-d after the crisis is resolved, and this itself will be effective in resolving the problem. Furthermore, the Ben Ish Hai adds, reciting this prayer on Shabbat has the effect of preventing troubles and hardship. For this reason, we first recite "Az Yashir" – the song sung by Beneh Yisrael upon being rescued from the Egyptians – to praise G-d for saving us from past crises – and we then recite "Nishmat," which helps to prevent crises in the future.
These sources should alert us to the special significance of this prayer text, and the need to recite it slowly and with proper concentration. Indeed, we hire Hazanim to sing it in a special and beautiful way, because of the special power of this text. By reciting "Nishmat" the proper way, we can enjoy the unique spiritual benefits it offers us.
