Short answers, pulled from the story.
The siege lasted from the 4th of June 1796 to the 2nd of February 1797, a period of roughly eight months, with a short break when the French were forced to lift the siege during the first Austrian relief attempt.
Mantua was nearly surrounded by water in 1796 , a large lake formed by the Mincio River bordered its north and east sides, and the city connected to its suburbs only by causeways. This made a direct assault very difficult. After the French lost their siege cannon in August 1796, Bonaparte was left with no option but to blockade the city rather than breach its walls.
Austria made four separate relief attempts. The first was led by Wurmser in late July 1796; the second by Wurmser again in early September 1796; the third by Alvinczi in November 1796, culminating at the Battle of Arcole; and the fourth by Alvinczi in January 1797, ending with the French victory at Rivoli.
The Austrians reported 16,333 killed, wounded, or dead of disease during the siege. Historian David G. Chandler estimated that as many as 18,000 Austrians and 7,000 French died in total over the course of the operation. When Wurmser surrendered, only 16,000 Austrians were fit enough to march out under their own power.
In recognition of his determined defense, Wurmser was freed along with his staff and given an escort of 700 soldiers and 6 cannon. The rest of the garrison marched out with the honors of war and was paroled on condition of not fighting against France until exchanged.
The fall of Mantua, together with the heavy losses Austria suffered during the four relief attempts, left Austria without the means to continue the fight in Italy. Bonaparte quickly moved to eliminate the Papal States army at Faenza, then launched a final offensive toward Vienna in March 1797. Austria sued for peace that year.