I'm just answering this because nobody else seems to have. If I have time later I shall return with my sources.

The Rising was never actually intended to succeed - the rebels were outmanned and outgunned (especially after Roger Casement was arrested trying to smuggle German guns into the country.) Rather the leaders expected to be crushed but, by doing so, to inspire others to rebel as well. This was at least partially inspired by the anticipated introduction of Home Rule (limited devolved government) after the Great War was over. Most extreme republican groups were by this stage quite isolated and the Irish Parliamentary Party (Home Rule Party) was the leading voice of nationalism in the country. The purpose then of the Rising was to evoke a sea change in opinion and try to get people to commit to complete independence.

This should not be surprising when we look at what the rebels selected as a focal point for the rebellion - the GPO. The GPO is in the middle of O'Connell surrounded on all sides by tall buildings from which snipers can shoot. It confers no real tactical advantage and is not easily defended (unlike say Dublin Castle or Trinity College or even the Four Courts). But it did have a number of other important characteristics:

1. Its location in the middle of the main street in Dublin ensured maximum chaos and disruption.

2. Its function as a post office meant that it was a relatively busy site (again ensuring maximum chaos and disruption)

3. The British were likely to shell the location. As it was in the middle of the city this was likely to cause civilian loss of life and certain to cause property damage - both of which would hopefully be held against the British.

4. Other locations taken around the city were also largely symbolic (like Stephen's Green) and conferred little tactical advantage.

5. Areas where they thought they would meet with resistance like Trinity College and Dublin Castle were initially avoided (although attempts were made on both when it was seen how lightly defended they were)

As to why sentiment changed in their favour there are a number of factors. Certainly to begin with most people were disgusted by the Rising and saw it as a betrayal of those soldiers off fighting Germany in Europe. There are reports of rebels being given an earful by natives of the city as they went about their business (especially around the Stephen's Green area). The most prominent factor was the British administration's execution of the Rising leaders and their institution of martial law. Propaganda-wise this was a coup for the republicans, especially when James Connolly who had been injured in the fighting was propped up on a chair before being shot. All of this served to portray the British as the dictators the rebels claimed they were. This was especially problematic when reports emerged of the "heroic" behaviour of the leaders before their execution (for example Joseph Mary Plunkett marrying his childhood sweetheart hours before his execution).

Also important was the fact that the rebels who were not executed were interned along with hundreds of suspected IRB members (many of whom would have been innocent). By sending so many hitherto isolated republicans to internment camps (notably Frongoch) they allowed them to intermingle. By spending so much time with like-minded men they were reaffirmed in their beliefs and radicalised while they were also able to establish new leaders and organisational structures (Frongoch saw the rise of Michael Collins) to use on their release in December 1916.

The final major factor then was the British response to the new IRA movement. Many were still sceptical of the republican movement and their initial attacks on the policeforce (who were all Irishmen themselves) didn't earn them many admirers. Churchill's decision to introduce the Black and Tans who were pretty much as brutal as their reputation was a huge mistake. It turned many Irishmen irrevocably against the crown and made republicans of many moderate nationalists.

Really the goals of Easter Rising being achieved had less to do with their own actions then and more to do with incredible British mismanagement of the situation. As a disclaimer I think 1916 was abhorrent and I am deeply ashamed of how much my country makes of event we should all try to forget. You should probably keep that bias of mine in mind when reading this.