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One reason there are fairy-tale German castles is that King Ludwig of Bavaria built them in the 1800s to be fancy and elegant, but castles too. Neuschwanstein is the classic example.
This just exacerbates the problem of separating castles - defensive war machines - from palaces like Versailles or palace-like chateaux. The Louvre started as a serious castle defending Paris - you can see it in The Très Riches Heures of the Duc de Berry - only to be defanged and prettified especially by Louis XIV and Napoleon I.
In England, the Plantagenets lived in castles.
I think your teacher has confused life "on progress" with a few instances of royal fugitives. It used to be normal to change residence every few months because a royal court had a lot of mouths to feed and chamber pots to empty. Rather than hauling food frrom all over the country, they would travel to the food, and a new clean castle. This also let them keep an eye on the country. This was done through the Renaissance.
I would say "fleeing" is a rather strong word. At least speaking for the Carolingians, castles and royal estates would be surrounded by royally owned, peasant-worked farmland, with small industrial centers for milling, smithing, weaving, and so on. Charlemagne's crew would move from residence to residence for a variety of reasons, but here are the most important ones:
Royalty traveled with large bands of family, nobles, advisers, guards, and servants. They would require spacious, well-stocked lodgings. The end of a stay coincided with the emptying of the cellar, as the party packed up and moved on to the next estate.
Moving around the territory was key in reaching the various lords protecting and administering different areas; Charlemagne's reign was immensely successful, in part because he was always making visits to the people helping run his kingdom, hearing their grievances and bestowing them with gifts and good company.
Going back far enough, it's difficult to think of certain accommodations as "comfortable" from a modern POV, but the wealthy would have had furs and fabrics with which to decorate and soften the castle.
A castle should impress and entertain. Charlemagne's favorite residence at Aachen had a hot spring that could fit 1,000 guests, an ornate chapel, and other lavish attachments.
For firsthand accounts, letters, and itemized lists of the castle stock, Paul Dutton's Carolignian Civilization is a fun read. Roger Collins' Early Medieval Europe is good for more general, encompassing information about movement, relations, and war.
Okay, this is a bit of a tricky one to answer because English doesn't have two words to differentiate between defensive castles like Eltz and purely residential palace type structures like Neuschwanstein. German calls the first "Burg" and the second "Schloss", and both tend to be translated as "castle".
actual European castles were nothing like the fancy, elaborate buildings that you see in Disney movies.
If you're talking about the second type, they pretty much are. Most of the castles of the Loire Valley for example tend to be luxurious palaces. On one end you have Château de Sully-sur-Loire which is an example of a defensible castle that was converted and had wings added to turn it into a more comfortable palace type castle. And then on the other end there's Château de Chambord where the castle-like towers and layout are entirely cosmetic and the building was never meant to be defended. It was used as a hunting lodge by François I and was only meant to entertain hunting parties for short times.
The classic Disney castle Neuschwanstein in Germany is a 19th century romantic fantasy project by King Ludwig II of Bavaria. If it had been finished, it would have been a luxurious mediaeval legend and Wagner themed palace. As it is now, only two floors are mostly finished, but they'll have all mod cons for the time.
The first type of castle, the defensive structure, tends to be a bit more functional on the inside. But there are so many variants here, it's almost impossible to cover all. It also really depends as well on how long it was occupied. What you see a lot with castles is that the owners convert it to make it more comfortable and fashionable over time. Even for functional castles you see the owners move out of the main keep, and build a big hall within the castle's curtain walls. Trim castle in Meath, Ireland is a good example where the owners moved out of the keep in the 14th century and lived in a great hall within the curtain walls. The keep would still be in use and house the garrison, but the lord had moved to more comfortable quarters. If the castle was still occupied after it lost its importance as a defensive structure, the changes could be quite drastic; Kilkenny Castle in Ireland for example demolished one wall of the castle to open up the courtyard to the gardens and turned the castle into a mansion-like structure in the 19th century. The site I link to has an excellent overview of all the changes made over the years in the "tour of the castle" sections which explains both the defensive and domestic changes made to the castle over time.
If the main castle just wasn't big enough, the owner wanted something more modern, or it couldn't be converted, you get something like Ross Castle in Killarney, Ireland where the main keep was left to its own devices after 1688 and the owners built an mansion on the castle's grounds.
She said that those castles were created primarily to keep royalty safe for a while as they were fleeing from castle to castle. In other words, they were only meant as temporary lodging and never intended to be comfortable enough to live in for extended periods of time.
That's pretty much wrong on all accounts. It could have been the case in extreme circumstances that royals had to flee from castle to castle to keep out of the clutches of a pursuing army, but that would not be their primary function. Generally they're build to control and defend something important. It could be a trade route, a border, a strategically important location that would deny an enemy access, etc. Most European castles would be owned by a local noble who would live there permanently. It kind of sounds like your teacher is mixing up a couple of things here, mainly that some early/mid mediaeval royal courts were not exactly fixed to one specific palace or castle, but tended to move around a lot. The court would travel from castle to castle (or monasteries), to ensure the nobles stayed loyal, to get a first hand overview of what was going on in the country, to show favour to nobles, and also sometimes to subtly punish a noble by staying so long, that it would drain that noble's coffers. The itinerant court or king was fairly common in mediaeval Europe, and for example the Ottonian and early Salian courts were pretty much always on the move. ( Every Inch a King: Comparative Studies on Kings and Kingship in the Ancient and Medieval Worlds - p 303 ) .
Why would anyone need to flee from a safe, giant stone building surrounded by a moat?
Because they weren't necessarily safe. A castle is only safe if the enemy doesn't have an overwhelming numbers in its favour that makes storming the castle an option, you as the defender have plenty of food and water to survive on for potentially a very long time, there are no big construction projects being executed by the attacker like mines and/or siege machines, or you have some hope to get out of this situation in some form or another. What most people forget is that while you're being besieged, you're effectively imprisoned while the enemy can move around and could be doing all sort of things to your lands like plunder your crops, steal everything that's not bolted down, etc.. Also you might need to be out there to rally your supporters and raise an army, find cash to pay for your campaigns, or make sure no one decides that this would be a good time to rebel.
And weighing up those risks, you might decide that staying mobile and fleeing that fancy castle is your best option, before the enemy closes off all possible supply and escape routes.
Most European castles were residences for a single knight (seigneur in French) and his family and servants.
They were small castles. They were built (usually) to defend a village and the village people.
There are thousands of these castles (or their remains) across Europe.
Royal castles and military castles were generally much larger and grander, and are more commonly depicted in modern representations and photographs, because they are more impressive.
Royalty (and very high nobility) often moved through the year from one castle to another. This was not because they were "fleeing" from one strong point to another. It was for two reasons:
They were keen to be visible in all parts of their domain and to dispense justice and resolve local disputes.
The cost of supporting the Royal Court could not be borne for very long by a single castle and its dependent lands, so the court would move from castle to castle to spread the economic burden.
Castles were generally very defensible. It was difficult to assault them (until the advent of mobile artillery siege trains). Castles were designed to be able to hold out for a long time with a small garrison against vastly superior numbers. They would hope they could hold out long enough for the enemy to give up, or for allied help to arrive.
The purpose of most castles was to defend a particular (fairly small) territory (the territory of one village and one knight).
The purpose of military and royal castles was to defend larger territories, and make it more difficult for invading forces to penetrate without leaving multiple castles in their rear, which might disrupt their supply lines and make life difficult for them.
The purpose of castles was not really to provide a chain of refuges for royalty to flee from one to another in order to keep themselves safe.
How impressive and "elegant" the interiors of castles were varied a lot depending on the period and the type of castle. Castles in the early Medieval period, when (in Northern Europe) they were often "Motte and Bailey castles", made of earth and wood, would not generally have vastly spectacular interiors. Neither would the small castles built to protect every village in the South of France. Military castles were functional, and their interiors might not be lavish. Royal castles, which were often the residence of Kings or High Nobility might well be extremely lavishly decorated (for their era).