At the time, manliness did not automatically mean an inability to emote. Manliness in the 1700s and early 1800s had two components - the plain-speaking, patriotic, independent man, and the paternal, emotional family man, which were combined in the ideal. This was particularly true for military men, who were expected to be able to give their all for their country and for their family. In this context, men were expected to show their emotion in public, and crying was an important part of this. Nelson's death must be understood in this context. While some of the stories will have been exaggerated, and romanticised to fit this concept of manliness, men would have cried at his death. He was an incredibly well-respected figure, and loved by his men. To lose him would have been the sort of event that Georgian society would have expected a sailor, no matter how hardened, to weep at.

Source:

Weeping Sailors: British Manliness 1760-1860, Joanne Bailey, http://www.britishnavalhistory.com/weeping-sailors-british-manliness-1760-1860/

Yes, it is quite true. It's also true that Nelson asked his flag captain, Thomas Hardy, "Kiss me, Hardy" as he lay dying. This quote was often later bowdlerized or misquoted/misinterpreted as Nelson saying "Kismet, Hardy" (Kismet being a Turkish or Arabic word meaning "destiny"). As /u/thefourthmaninaboat mentions, public displays of emotion were quite common at the time, and weeping at the loss of a comrade would not at all have been unusual.