Did it help change the plight of the slaves? Or is it a narrative we read to look back and say gee yea slavery was really bad.
Something to be mindful of is that Equiano's book stands out in part because it has survived in widespread circulation long after publication; it was one of many dozens of narratives produced by victims of slavery at the time. Numerous other texts - ranging from lengthy books to brief pamphlets - were published with slave autobiographies, though the authenticity and accuracy of many of them is questionable. Furthermore, Equiano's book is not quite a true slave narrative; the majority of it actually deals with his life after slavery and not just with the experience of slavery itself, unlike most other published work which concerned itself only with highlighting the moral decadency of slaveholding practices.
Certainly, his book stands out in the sense that it was a remarkable success. It was published in 1789 and enjoyed remarkably high sales until 1794, was translated into multiple languages and was reprinted multiple times - on average in that period, nearly two new editions a year were published. Much of this long-lasting success is attributable to the fact that Equiano aggressively marketed his story with an extensive tour of the UK, where he both sold the book and agitated for anti-slavery mobilisation among all who would listen; contemporary news stories testify to the persuasive strength of character and intellect he demonstrated in person. Though he had his fair share of contemporary critics too, in that sense the book made a major contribution in enabling Equiano to disseminate his anti-slavery ideology to a wide audience and expose more people to the movement, and the overall reception was resoundingly positive. Each new edition of the book was tailored and modified by its author to try and maximise its appeal to a wide audience.
Ultimately though, its success was made possible because it was published at a time when susceptibility to abolitionist sentiment in the UK was strong; the French Revolution had just begun and Thomas Paine's The Rights of Man was on the cusp of publication. Equiano's book made compelling moral and economic arguments for the abolition of slavery and enjoyed widespread support from prominent abolitionists, who were becoming a serious force in British politics for the first time; he was best received in parts of the UK where there was already an appetite for social reform. Ultimately though this period of fervour wouldn't last - by 1794, the UK government was keen to stamp out social reform movements.
Campaigns like abolition could not be easily disassociated from campaigns for other social change like electoral reform which presented a tangible threat to the polite and social elite, and revolutionary France - with whom Britain was now at war - had abolished slavery in 1794, creating or reinforcing a dangerous link in the minds of the establishment between British abolitionism and anti-establishment sentiment. From May 1794, new anti-sedition measures were used to target political and social radicals who may have posed a threat to the British establishment - in the trial of one prominent radical activist, a letter written to him by Equiano was taken by the authorities of possible evidence of his treasonous character (though he was acquitted). Around the same time, Equiano had exhausted his list of major speaking destinations and earned enough money for himself to retire in comfort. From 1794 and especially 1797, interest in the book went into decline. In the United States, although it had at least one major reprint in 1837, it would be quickly surpassed in contemporary and historic significance by works like Frederick Douglass's Narrative and Uncle Tom's Cabin and spent many decades without a new edition in circulation.
Equiano himself died in 1797 but the slave trade would persist in Britain for another decade; amelioration (measures aimed at improving slave living and working conditions) imposed by the British government would not come in any meaningful way until 1823. Slavery itself would not end by legislation in British colonies until 1833 and not fully in practice until 1838. In that sense, the book did little to achieve any kind of meaningful reform directly. It did however help to give the anti-slavery movement an authentic and real African voice, and an icon who could be an example both of the devastating impact of slavery and what it was possible for Africans and their descendants to achieve as free men and women in society. Equiano's own efforts at promoting the book helped to stimulate and organise anti-slavery activity across the UK and expose many thousands of people to the abolitionist movement first hand. In that sense, it was a success at agitating for change.
If you're interested in reading more about this, I would recommend Vincent Carretta's Equiano the African: Biography of a Self Made Man (University of Georgia Press, 2005), which is an exploration of his life and character, formed through a critical analysis and contextualisation of the original biography.