To give a simple answer to your question; Monaco survived thanks to its status as a protectorate of larger countries that others didn't want to mess with, mostly France though, at the time of Italy's unification, Sardinia.

Monaco had retained sovereignty throughout the medieval period, but had always done so through alliance with a larger nation, mostly France, with a number of Monégasque lords holding military office in the kingdom despite their independence; Rainier II, for example, was Admiral of Languedoc and fought with the French at the Battle of Poitiers, and was captured by the English while escorting French merchant ships in the English Channel.

Monaco remained a subject of France until 1523, when Augustine Grimaldi — regent of Monaco to his 9-month-old nephew Lord Honoré I — severed Monaco's relationship with France and swore Monaco's allegiance to Spain, allowing Monaco to focus on advancing their own position under the military protection of Spain. This allegiance saw Monaco subsequently losing some of their French holdings in Savoy.

Of course, this wasn't always a beneficial relationship for Monaco, especially in terms of tribute paid to Spain for their protection; Spanish troops also occupied the country from 1604 to 1614, during the minority rule of Honoré II, and Spain wouldn't actually recognise Honoré as a sovereign prince until 1633. In his adulthood, Honoré II became a critic of Spanish rule over Monaco, and in 1641 ended it completely with the signing of the Treaty of Péronne with Louis XIII of France, which established Monaco as a protectorate of France while guaranteeing the position of Honoré and his successors as sovereign rulers over Monaco. This switch of allegiance saw Monaco lose possessions in Spain and France, though Honoré was made marquis of Les Baux and Duke of Valentinois by Louis XIII for his trouble; Monaco's rule over the cities of Menton and Roquebrune was also re-established.

Monaco remained a protectorate of France until 1793, when it was occupied during the Revolution and incorporated into the French département of Alpes-Maritimes. Monaco's sovereignty was re-established in 1814, though in 1815 it was made a protectorate of the Kingdom of Sardinia. It's position as protectorate of Sardinia allowed it to remain independent during the unification of Italy, which was itself an enterprise of the Kingdom of Sardinia, which in 1861 became the Kingdom of Italy.

The country seems to have narrowly avoided incorporation into the Kingdom of Italy when in 1860 the entirety of Sardinian Nice and Savoy were ceded to France, and Monaco was made an independent protectorate of the French instead. This secured French military protection for Monaco, but saw France annex the cities of Menton and Roquebrune, which accounted for 95% of Monaco's territory, in return. Monaco has largely remained under the military protection of France ever since.

As an interesting sidenote about Monaco's status as a tax haven; there was precedent set for this as early as the 13th century, when settlers from Genoa established a military fortification and surrounding settlement on the Rock of Monaco, and offered tax exemptions to new settlers (though this is not to say that it was always a tax haven, of course).

I hope this answers your question, though I'm more than happy to cobble together some answers if you've got any follow-ups. This sort of period isn't really in my usual field of expertise, but the existence of microstates is one I've delved into independently out of sheer curiosity.

Sources:

• Christiaan Klieger, P. 2012, The Microstates of Europe: Designer Nations in a Post-Modern World
• 'Principality and Diocese of Monaco', The Catholic Encyclopedia
• Pemberton, H. 1867, The History of Monaco: Past and Present

Unfortunately sources on Monaco are few and far between, but we make do.

I see there are several other answers here, so I'm not going to bore you with too much pedantry, but an idea worth exploring is the way sovereignty was formerly understood to work in Europe.

While most will point to the Peace of Westphalia and the end of the Thirty Years War as the watershed marking the definitive emergence of contemporary notions of political sovereignty, the actual process by which sovereign status was achieved differed in various parts of Europe (in other words, the Treaty of Westphalia was not an "Abracadabra" moment). In many respects, contemporary Europe is much more defined by the Congress of Vienna and the end of the Napoleonic era than it is by "Westphalian Sovereignty," ultimately meaning that our notions of what constitutes independence and sovereignty are much more recent than we might expect.

So while it might be strange to do so, prior to the Congress of Vienna it is nonetheless possible to encounter political actors, units, and subdivisions that do not fit into any well-defined constitutional hierarchy as we would expect in an age defined by nation-states and increasing formalization of relations between sovereign powers.

Monaco is one such polity, and being a "Protectorate" that could shift allegiances between great powers (as has been pointed out elsewhere on this thread) would not be entirely unique thing. As an example, harbors along the Tuscan coast (notably Piombino) also came under the "protection" of the Spanish Empire much like Monaco did, without explicitly becoming independent of their "Sovereign" (in this case, the Grand Duchy of Tuscany). Another manifestation of overlapping and ill-defined divisions of authority saw aristocratic privileges and holdovers from feudal investiture create semi autonomous territories within well-defined borders: an example of this is the "Stato Borromeo," a highly autonomous subdivision of Spanish Lombardy.

Monaco (and its peers) are notable to us because they escaped consolidating forces which eliminated semi-autonomous territories elsewhere. In Monaco's case, this escape came with a fair bit of luck: When the Kingdom of Sardinia-Piedmont (itself a haphazard invention of the Congress of Vienna) ceded the counties of Nice and Savoy to Napoleon III's French Empire in exchange for military aid in the struggle to unify Italy, Prince Charles III of Monaco pulled off an astounding coup in seizing that opportunity to formalize his fief's autonomy and extract Monaco from the cession. If some of the semi-autonomous sovereigns mentioned above, like the Prince Borromeo in Lombardy or the Prince of Piombino in Tuscany, had been presented with a similar opportunity perhaps they would have siezed it as well. As fate would have it, they couldn't and didn't.

Monaco is a bit of a hobby area of mine, and I wrote a few posts which might interest you:

• A generic post narrating the history of Monaco, which looking back is a little lacking in examining social and political drivers
• This post on Monaco's modern economy