Francis II dissolved the Holy Roman Empire only a few days after the formation of the Confederation of the Rhine by Napoleon, and the announcement from many of the smaller German states that they were leaving the Holy Roman Empire in favour of the Confederation of the Rhine. Napoleon had actually issued an ultimatum demanding that Francis step down, and while that was definitely a factor, there were other reasons for the fall of the Holy Roman Empire.

With the utter humiliation of Austria by Napoleon, and the organization of the Confederation of the Rhine by Napoleon many of the Smaller German states left the Holy Roman Empire. With the rise of Prussia, Germany had really been split into three spheres of influence, Prussia, Austria, and the rest of Germany. Many of the smaller German states threw their lot in with Napoleon and joined the Confederation of the Rhine for numerous reasons, one they feared Napoleon and his armies, secondly they figured that by accepting Napoleon's protection in the Confederation they might better position themselves for the future with or without Napoleon; basically they hoped build themselves up and prepare for the future. Finally, the reason many of the smaller German states joined the Holy Roman Empire and still participated was because it provided protection from bigger states, Austria had basically made the title of Holy Roman Emperor into a hereditary title, so any member of the Holy Roman Empire was basically guaranteed the protection (in theory anyways) of Austria. Napoleon showed that the protection offered by the Holy Roman Empire was not clearly all it was cracked up to be. Furthermore, after the treaty of Luneville states in the Empire and outside began to just annex the various small German states. Austria, Prussia, France, Bavaria, Baden, Wurrtemburg all gained substantial territory, many of the smaller German states were wiped of the map. It really undermined the structure of the Empire that member states and the leader of the Empire could just at any moment rearrange the territory of the Empire.

Beyond that though, the Holy Roman Empire had simply just become stagnant, the Peace of Westphalia in 1648 that ended the Thirty Years War took away the ability of the Emperor to decide on religious matters in Germany which was a major part of the original structure of the Empire. There had been attempts to reform the Empire since around 1790, specifically with regards to secularization and federation. This had promoted some, such as Karl Theodor Anton Maria von Dalberg, to think the Reich was collapsing and thus they tried to save it. Dalberg was a high ranking official in the Holy Roman Empire and was the head of one of the Archbishopric states (small little religious states that were in the Empire). Basically he saw that no one in the Empire, including the Emperor had the will to actually go through with the reforms, so he turned to Napoleon to save the Empire, by offering members of Napoleon's family high ranking offices in the Empire. Napoleon wanted none of this, he had seen the issues that being Emperor had caused Austria, and he preferred to keep his little system of Client States intact rather going and getting involved in the decaying Empire. Napoleon planned to just let the Empire die off. Dalberg ended up jumping ship and joining the Confederation of the Rhine, seeing it as the successor to the Holy Roman Empire. He tried to do what he had tried to do in the Holy Roman Empire and reform the Confederation of the Rhine, but everyone kinda just ignored him. The German princes who were leaving the Empire said that

They had effectively been abandoned by both major German powers; ‘in vain’, they declared, ‘did one look for Germany in the heart of the German polity’; those whose lands lay nearest to France had had no choice but to seek peace with France; now they were all resolved to seek the protection of the Emperor of France."

In the closing days of the Reich there was some attempt to mould it to fit the new political situation. Bavaria proposed that the Empire turn into a permanent German federal state of sorts, with a figurehead Emperor chosen by election, the Reichstag would be transformed into a permanent body of representatives. Prussia tried to build their own quasi Empire in the North of Germany, both suggestions didn't pan out. There were reports from Austrian officials saying that the Holy Roman Empire had served its purpose and was no longer worth maintaining in the face of French pressure. For example one was released by an Austrian politician and official in the Holy Roman Empire named Joseph Hass:

Foreign power and politics’ had now achieved what the unruly spirit of the German princes and their mistaken sense of freedom had failed to do over centuries: the unity of the German Reich was destroyed. ‘The German people no longer constitute a state’, he wrote, ‘not because of the desire for independence of its Estates, but rather in order to impose a foreign yoke all the more rigorously.’ Virtually nothing was left of the system of the Reich; the emperor now only had authority in the Archchancellor’s lands, in Western Pomerania, Mecklenburg, and the Hanseatic Cities—‘for as long as these lands retain their present rulers’.

Francis decided to relinquish the Crown and end the Empire, but only after making sure that he was confirmed as the hereditary Emperor of Austria and all the lands that went with it. Francis didn't reference the French ultimatum, simply saying that he had tried to fulfill his duties as Emperor. He stated that since he had lost the confidence of the Electors and Princes he would abdicate and release all servants of the Reich from their duties. He said:

‘for as long as we can live up to the confidence placed in us by the Electors, princes and Estates and other members of the Reich, and for as long as we are able to satisfy the obligations we have assumed’.

we hereby decree that we regard the bond which until now tied us to the state of the Reich as dissolved’.

In the mind of Francis, it was better to end the Empire and confirm his own power as Emperor of Austria, than risk the Empire falling into the hands of Napoleon, or one of the Electors. An Empire dominated by Napoleon was an Empire Francis did not want to be apart of.

My main source for this would be volume II of the book "Germany and the Holy Roman Empire" by Joachim Whaley. But the book is very long and isn't the most riveting read, so if you're looking for something to read on the matter, I'd advise looking for the book "The Holy Roman Empire 1495-1806" by Peter H. Wilson