AH! The Stuart Monarchial era of British History is one filled with so many Ups and downs, I hope I can answer your question.
I will try my best to not give the extremely long winded evidence based answer, and Ill leave it up to you to ask for clarification if you need. Charles I is the rock that caused a massive ripple throughout English history. His family, the Stuarts, have an intense history starting all the way back with his Grandmother, the infamous Mary Queen of Scots, and ending finally with Queen Anne, the last Stuart Monarch (Fun Fact: The Jacobite rebellion of 1745 was perpetrated by Charles' later Descendent, Bonnie Prince Charlie).
But the Key element of your question that needs a small time line to understand the answer fully is "Why did the English stick with the Monarchy". Well short answer first, they didn't. Following the death of Charles I in 1649, England was declared a Republic under the leadership of One (and one of the funniest/worst leaders in all of European History) Oliver Cromwell. Under his leadership and the growing Puritan majority within the House of Commons, Oliver Cromwell ushered in a reign of about 12 years of English History known as the Commonwealth period. It was during the Commonwealth Period (1649 - 1659) that England existed as a republic devoid of a monarch. The House of Commons was the key stronghold and bastion of this new Republic, Ironically enough though, not the progressive positive entity that everyone believed would come into power following the chaotic rule of Charles I (The guy literally lead England's Gentry into Two civil wars against the House of Commons aka the Parliamentarians after having the House of Commons dissolved over 5 different times).
It was under the Reign of Cromwell that he created his "Rump Parliament" in which he fired anyone from the House of Commons that did not support his/the Puritans social agenda. Under the Rump Parliament and Oliver Cromwell, a huge Puritanical social revolution occurred that flew in the face of many things that defined English as a People and a culture. Things outlawed in many Puritan communities (but not all) under the reign of Cromwell included....
This list doesn't even include his treatment of Catholics in Ireland...
...But at least Cromwell allowed the return of the Jewish Population back to England that had been banished since 1290...
Now, Cromwell and his rule would eventually progress resulting in a Military Dictatorship under the entity known as the Protectorate (most of the bans listed above happened under the Protectorate Period), where he himself would be proclaimed Lord Protector (again, Ironically becoming more of an Absolute Monarch than Charles I ever was). And for even further Clarification, this was all happening while Charles I's son Prince Charles (Later Charles II) was fighting against Cromwell in the Third English Civil war, had been declared King of Scotland, and was trying to retake the throne. However, even though Cromwell had a large support from his New Model Army, he would die in 1658 with a large amount of debt, some pretty upset britts, and a power vacuum that was filled by his son Richard Cromwell. Richard was unpopular and not as effective of a leader as his father, so the Houses of Parliament instead decided to Enter into negotiations to return a Monarch to the Throne.
Due to the distaste for the Social Revolution by the Puritans and the hatred of the Protectorate, the Parliament brought back Charles II son of Charles I as the New Monarch. This period was called the "Restoration Period". Charles II was a widely popular Monarch. Known colloquially as the Merry Monarch or what I call him in Class "The King that brought back Partying". He overturned the Cromwellian Reforms and brought back everything from Promiscuity to Christmas. So much of the answer to your question lies in the transitional period known as the Commonwealth, and the distaste for Military Dictatorship brought with it. Ultimately British people wanted their monarchy back after getting a taste of a Republic being lead by an intensely divisive figure like Cromwell. Now if you keep going down the rabbit hole, you'll get into a lot of fun stories about Charles II, his love for Tennis and women that weren't his wife, eventually culminating in the Glorious Revolution under his brother James II.
I hope this answered your question, sorry if it was a lil long winded.
Well, if you asked someone in 1660, they'd likely say that the ending of Cromwell's anti-monarchical interregnum was divinely inspired; that the Protectorate was a mere aberration in centuries of glorious monarchy, so much so that the official date of Charles II's reign began literally on the day that Charles I was executed, even though in reality Charles was executed 11 years before in 1649. More recent historiography, however, emphasizes the popular nature of the Restoration; the way that it sprung up from popular political sentiment that both Cavaliers and Parliamentarians mobilized in the 1640s and 1650s.
The English Civil War, depending on who you ask, is either an ascendant commercial gentry class taking power against an unstable nobility in the Marxist interpretation, a process of democratization in Britain's long road to parliamentarianism that started with the Magna Carta in the British nationalist/Whiggish interpretation, or, the generally accepted scholarly interpretation today, a confluence of events that happened all just at the right time, that was either more or less avoidable had Charles I, who was more in the mold of an absolutist monarch which didn't fly with Parliament, and certain members of parliament taken different steps. But, alas, they didn't, and starting with Charles' attempts to standardize Anglican practice in the Scottish Presbyterian church, and despite the fact that many people expressly didn't want Civil War, two major factions formed: those in favor of the monarchy, and those who opposed it, led by the zealous Oliver Cromwell, a gentleman known for his piety and military prowess. Cromwell's New Model Army defeated the royalists and executed Charles I for treason in 1649.
A major part of the Civil War was the explosion of popular politics within it; movable type and increasing literacy meant that more people were able to read and publish pamphlets. For the first time, despite the existence of laws against slanderous libel, even monarchs' sexuality was fodder for the printing press, discussed at taverns and, by the 1660s, coffeehouses. Cromwell faced the tough balancing act in his reign of both appropriating monarchical iconography and symbolism and distancing himself away from noble ideals in line with his politics. Cromwell's desire for "plainness," in which he subverted gentry ideals without rejecting them outright, encouraged satire against him and his wife Elizabeth that could be absolutely biting. Anyway, this explosion of literature has captured the attention of historians seeking to do "history from below," and those historians have added to the traditional causes of the collapse of the Puritan regime - the death of Oliver and the ascendency of his less capable son Richard, a rise in taxes from heavy military spending, Puritanical rule being unpopular with a mostly Anglican population, a sense of disarray - with an understanding of the ways that it was the Cromwells' loss of legitimacy both in the eyes of the nobility and a wider swath of English society who voiced their opinions that caused the collapse of the Protectorate. And the earlier years of the 1650s had constrained the forces of militarism and religious fervor; by the latest years of the decade, there were few limits to a soldier's power. So while Charles' contemporaries perceived the event as a stroke of God's favor, it was immediately a result of popular protests in 1659 against an increasingly strident Puritan regime. To what extent Charles' revolution was revolutionary is also an increasingly debated question in the historiography; the extent to which Charles's government looked like his father's or Cromwell's is a fertile field for scholarship.
To some extent, though, British citizens "sticking with the monarchy" each wanted something different. Anglicans generally wanted a movement away from the Puritan hardline politics of the parliaments of the 1650s; Catholics saw Charles' wife, a Catholic from Portugal, as a potential ally who could sway Charles towards sympathy; others saw in the Declaration of Breda, which was Charles' promises about his potential rule, either a desire for absolutist or limited monarchy, based on what one's preference was (this was the time of one of the most notable works of political theory, Hobbes' Leviathan, which claims that the only way to avoid humanity's natural state of crisis and war is to have an all-powerful king; Hobbes' social context was steeped in the crisis and war of the 1650s). If you were angry at the Protectorate, you could probably find something to like about Charles enough to agitate for him - he was, in the words of Tim Harris, "all things to all people." Charles aligned himself as the "merry monarch" who promised a return to normalcy and to move away from the austere Puritan culture of Cromwell's rule. He most certainly embodied that spirit, with his 12 illegitimate children. A testament to the optimism associated with the early period of the restored monarchy, John Dryden even wrote a 1667 poem, Annus Mirabilis, about how the crises of the year 1666 such as the Great Fire of London were in fact divine intervention to help Charles' agenda through. The Stuart line would last until the Glorious Revolution in 1688, the most important date in any Whiggish history of England, and in any case Charles' popularity soured as his reign continued, but that's beyond the scope of the question (and somewhat beyond my area of expertise).
Sources:
Tim Harris, "What's New About the Restoration?" Albion 29:2 (Summer 1997): 187-222.
Laura Lunger Knoppers, "The Politics of Portraiture: Oliver Cromwell and the Plain Style," Renaissance Quarterly 51:4 (Winter 1998): 1282-1319.
Samuel Fullerton, "Fatal Adulteries: Sexual Politics in the English Revolution," Journal of English Studies 60:4 (June 2021): 1-29.
From Republic to Restoration : Legacies and Departures, ed. Janet Clare (Manchester: Manchester University Press, 2018).
Edit: I'm not sure to what extent you wanted to know about the history of the Civil War or the Restoration, so I've added a brief section to make the Civil War's history clearer. Hope I could condense a very messy 20 years of history, full of shifting alliances, religious fervor, and mass politics, in a way that does justice to the volume of great scholarship on the period and to the people whose lives we have the privilege of studying and interpreting.