Alrighty, so there are two important facts we need to note here when we consider this question: underscoring that it is derived and noting the history of malta's interaction with foreign powers and the role that plays in linguistic history.

First, Maltese is an arabic-derived language, and it should be noted that much of the Maltese vocabulary shows extensive exchange with Sicilian, Tuscan Italian, French, and much more recently (albeit to an ever growing degree since it's been made one of the official languages) English. Here's an example:

The Netherlands are il-pajjiżi baxxi which literally translates as (the)-(countries) (low) or The Low Countries. It's very similar to the french Les Pays-Bas as opposed to al-Balad al-Munkhafid (البلد المنخفض) in Arabic, which is very different, obviously. It's helpful to consider this, as it's not as if people are speaking anything that an Arabic-Speaker would come close to understanding, and though an Italic Language speaker may be thrown off by some of the structuring etc., closer examination shows a language that actually has quite a few similarities.

The next part is a bit more actual history, instead of just linguistic comparison. Much of this history can be found in A Concise History of Malta by Carmen Cassar if you'd so desire to look it up. I don't have the book on me at this moment writing this, and I'm drawing from what I learned out of that book, as well as (inevitably) the nebulous body of things I've read about Malta as I studied the Fatimid Caliphate, North African history (specifically the Ablaghid Emirate), and Franco-Turkish relations through the 16th and 17th centuries. The Islands that constitute Malta were originally Phoenician, then Roman, and then Byzantine land. Under the Byzantines, Malta, and the settlements on it, were on the outer edges of the empire and looked more like frontier towns than integral and deeply involved parts of the empire. There was one major town of note that had existed since the Phoenician times, Mdina, retaining it Phoenician name. The Island would then be taken over by Arab conquerors. Now most of the records of this are in Arabic, which I unfortunately cannot read. However, the basic gist of it is that there were a series of Aghlabid arab invasions into the island of Sicily, and Malta was a convenient stop along the way. This would leave the Island in ruin, and provide a perfect opportunity for Arab-Berber people to populate the Island. When the Fatimids came over, they had an easy time of taking over the Island, and keeping it away from Byzantine recapture, since the Arab-Berber inhabitants felt much more at ease with the Fatimids than the Romanesque Byzantines. They also would be much more receptive to the Siculo-Arabic language that would develop in the region and spoken by the people.

It was all eventually taken over by Normans. Though they were mainly trying to take over Sicily here, and Malta was more just a piece of land that came with the package (also because it was too close to Sicily not to take, unless they liked the idea of Arab incursion). They would more or less rule the Island from afar. It would pass onto other European powers, most notably the Spanish. On the Island of Sicily the Siculo-Arabic language was wiped out all together, whereas in Malta they kept speaking it. The Knights Hospitaller were given the Island and lived on the Islands more as a class of elite than actual inhabitants. There was a great deal of trade on the smaller scale with Sicily, interaction with French traders (who had a vested interest in the very French Knights of Malta), and with the Italians that would later impose themselves onto the land as well -- all this explains the Tuscan Italian, Sicilian, and French influences on the language's vocabulary and structure. There were a couple small towns built, and some intermixing most definitely went on, but it was mostly with the neighboring Sicilians. After the Arabs, the Maltese were cut off from Arabic culture effectively, and the language gained much more vocabulary from European languages as a result.

But that doesn't explain everything. So now why, after all these conquerors and Romance-Language conquerors did the Maltese still speak a language that is classified as semitic? Much of its lies with the way the conquerors interacted with the Island. They ruled from afar, rarely visiting. This is much like how the Norman conquerors of the Britain spoke French, but because they spoke it amongst their elite class and the Island didn't have much migration coming in from Normandy, the native language of English remained, though to was heavily influenced by french vocabulary. Much the same happened to Maltese -- it's vocabulary is very much so influenced, but it still remains its characteristic roots and remains its own unique language. To put it crudely, there was not that much care invested into the Maltese to try and change the language that they spoke. Malta post-byzantines became a strategic area, but not exactly a rich Island that merited them doing much to integrate the native population on a linguistic level.

Now, in the 16 century and going onward, there were attempts to actually teach the Maltese Arabic so that there could be proselytization to the Christians of the Levant. These were mostly massive failures, though they can be read about in an interesting paper by, once again, Carmel Cassar, entitled Malta and the study of Arabic in the sixteenth to the nineteenth centuries.This idea of using some familiarity with semitic language structure to preach and interact with christians in the levant gave ruling powers a reason to let the Maltese continue speaking their language as well. The Maltese were left to themselves for much of history, and the fact that their ruling powers were often distant, or French Speaking Knights that didn't bother with them much made their language an integral part of their identity politics, and it is still a source of pride for the Maltese today. There are later attempts by Mussolini to classify Maltese as an Italic language with further spurred this identity as well. The Maltese under the british were also then identified as a sort of last bastion of Phoenician culture (remember Mdina?) and this would play into the desire to keep the language as well. After independence from the British in the 20th century, the Maltese kept English as one of their official languages, and there has been a massive amount of english influence (essentially all new words into the language are just the English word with the proper maltese endings applied so that it works grammatically), and many have predicted some fall of the maltese language in favor of english for a while now -- however nearly the entire population still speaks Maltese (though also know english for the most part now too), so that theory's probably bunk.

I hope this helped give you some answers. It essentially comes down to knowing that there are some big changes to the language, but also that there were no real efforts to replace the language and some efforts to capitalize on it. And growing nationalism in the 20th century would help cement the language as well.

could ask in /r/linguistics as well.