Westernization: Adoption of Latin alphabet was part of an extensive westernization policy, like adoption of metric system, Gregorian calendar, Swiss civil code etc. Not all the Young Turks, but some of them -including Atatürk- believed Turkey had to be a part of Western Civilization.
Nationalism: In order to underline Turkish identity -rather than Muslim identity- a campaign was carried out to remove Arabic words from Turkish language. In some cases, words from earliest Turkic inscriptions -which were long forgotten- were found out and replaced Arabic words. That campaign was successful in some degree.
Laicism: It is argued that Atatürk was anti-religious and he aimed to undermine Islam in Turkey. Banning Arabic script was considered an attack to Quran itself by some Islamists and anti-Islamists, however many refuse this speculation.
Republic of Turkey was founded with some kind of revolution, and that was natural to refuse some heritage of the previous system to legitimize the new one. Ottoman Empire was accused to be Arabicised and Turkish language was "rescued" by the new regime.
Ottoman Turkish and vernacular Turkish weren't the same thing even when the Ottoman Empire was still around. Ottoman Turkish borrowed heavily from Arabic and Farsi for its vocabulary (I've seen some estimates suggesting over 80% of the entire vocabulary was borrowed from those two languages alone), so the fact that Ottoman Turkish was written in an abjad script wasn't really a problem for the ruling elite. Especially not after Ottoman Turkish, like Farsi, added letters to the basic Arabic script. (They'd do the same thing with the Latin alphabet, by the way.)
But Arabic script is a problem for vernacular Turkish. It's a language rife with vowels and Arabic script doesn't have them. It's a language that simply implies them, leaving it up to the reader to use context to figure it out on their own. If you're working with a pool of literate elite who probably had extensive private education, it's a non-issue. But when you're trying to increase your nation's overall literacy, as Ataturk was, you want a system that's relatively easy to teach. The broad reform that he aggressively pushed didn't do as well as he hoped, but it still tripled literacy rates.
But it's probably important to also recognize the political nature of these reforms. The fledgling Republic implemented a broad series of reforms, which Ataturk styled as revolutions, that were aimed at both establishing a Turkish identity and irrevocably distancing Turkey from the Ottoman Empire. It also didn't make terribly much sense to hamstring the influence of Islam (abolishing the caliphate, closing dervish lodges, abolishing the sharia courts). It was also part of Ataturk's general plan to transition Turkey into, at least in the eyes of Europe, as a "proper" European state. I wrote a bit more on that here, though that post largely deals with what Turkey did after his death and immediately prior to NATO. But since the Turkish government was following his example, it's still topical.
As a Turkish speaker who can also read the matbuu (i.e. printed) Ottoman script, I can say that adopting the modern Latin alphabet was one of the most important revolutions of Ataturk. As other posters said, the script used in the Ottoman Empire was terribly ill-suited to Turkish language. Modern Turkish alphabet is almost entirely a phonetic transcription, which means you are able to correctly read and pronounce the words you do not know. In the Ottoman script, this is not the case, which comes as a shocker to the Turks who study the Ottoman script for the first time.
It is also important to know that the Ottoman intellegentia was perfectly aware of the problems of trying to write the Turkish language with Arabic/Persian script for a long time. It was a topic discussed among the scholars. Adopting Latin alphabet was one of the solutions discussed long before Turkey was founded. There are some texts or words written in Latin alphabet by some scholars during the Ottoman Empire, although without any standards. Some Ottoman scholars proposed Greek or Armenian scripts to replace Arabic/Persian script as well. Therefore, using Latin alphabet to write Turkish was not a new idea in the newly founded Turkey. Ataturk made that decision and standardized the Turkish spelling, as well as standardizing the official Turkish dialect as the Istanbul Turkish.