(This is a bit of an obscure example but this is the most prominent one I've read about since I'm a Pakistani.) For example, the language Punjabi was first written in the Gurmukhi script, but during the later years of the mughals it started to get written in Arabic script and that's how it's still written today. Why and how did changes like these happen?
I can't answer the the specific example you chose, but Hebrew had that happen, and you might find it of interest.
The oldest examples of Hebrew are written in the Paleo-Hebrew alphabet, a script closely related to Phoenician. The oldest Hebrew inscriptions are invariably written in this script.
In the 6th century the region was conquered by the Babylonians, then the Persians. The Persians, despite being Persian, used Aramaic as the language of administration. Aramaic was a major language used by a variety of governments in this period, and as a consequence was a major Middle Eastern language (though it is now very nearly extinct). Aramaic was the prestige language, and over a period of centuries many diverse groups in the Middle East began to replace their native languages with Aramaic.
This process happened with Jews too, and Aramaic gradually replaced Hebrew. This process took centuries. But while Hebrew continued to be used Aramaic had a major influence--Hebrew switched to an Aramaic-derived script. This script is related to Paleo-Hebrew, and there's an exact 1:1 letter correspondence. That certainly made the switch simpler, since it affected only the ways letters looked, not the writing system itself.
Paleo-Hebrew still had some currency, but declined in usage. The last inscription in Paleo-Hebrew are coins from the Bar Kokhba revolt in 135CE, which is around the time Hebrew was definitely dead as a natively spoken vernacular language.
In this case no one bothered to say why they were making the switch. Literate people just gradually switched which sorts of letters they wrote. But, the social role of Aramaic is the key here--once scribes were using Aramaic to communicate with the government, they continued to use that script to write Hebrew. Or alternatively, with Aramaic script as the prestige script people used it because of that. Ultimately we know that Aramaic script was used for governmental administration and was the prestige language/script, but the exact dynamics of the switch aren't known exactly.