It was based on evidence from other countries that giving schools certain freedoms, including the freedom to compete for pupils (which LA maintained schools cannot do), led to better outcomes. Also, LA maintained schools have their budget top-sliced by the LA and many felt that they didn't need some of the services the LA provided and that they could get the services they did need at lower cost, freeing up more to spend on educating the pupils. That is still a motivation for schools to convert.
At the time Labour introduced academies, there were significant numbers of schools that had been failing to perform at an acceptable standard for a decade or more. Some LAs had significant concentrations of such schools, so it was clear that the LA was not doing anything effective to improve school performance. Removing such schools from LA control was therefore seen as essential to improve performance.
It was also believed that the academy model would give parents greater choice, with different schools offering different approaches. I'm not convinced this has actually happened to any significant degree.
Contrary to what @willingtolearn says, it was not about "filtering public money into wealthy people's pockets". It is a legal requirement that academies are run by charities. They may be constituted as a company (many charities are), but it is a company limited by guarantee. That means it doesn't have shareholders and no-one can own the charity. It also means that the charity cannot distribute its profits to its members and that the majority of trustees (the people who control the charity) are unpaid. Where trustees are paid, they cannot be paid simply for being a trustee and, if challenged, the charity must be able to show that they are receiving goods and/or services to at least the value of the amount paid. These controls also apply to paying close relatives, business partners and businesses connected to trustees. Of course, none of this is an absolute guarantee against fraud, but fraud happens at LA maintained schools as well.
What the FullFact article misses is that evidence from other countries is that academy-style schools drag up standards generally - that LA maintained schools (or their equivalent) perform better where academy-style schools exist. The fact that the UK has moved up the PISA rankings significantly would be seen by supporters of academies as evidence that it is working. The fact that England has moved up significantly whilst Scotland (which does not have academies) has gone down would also be seen as evidence in favour of academies.
Another point re the FullFact article is that its data is from 2016, at which time a lot of academies were forced conversions of schools judged to be failing. It might, therefore, have been instructive to see how much such schools had improved compared to LA maintained schools.
My personal view is that it is clear the UK's education system has improved relative to other countries over the last decade. I have no idea how much, if any, of that improvement is down to the academy programme. However, it is clear that academies are here to stay.