The most significant thing that happened outside the 20 year rule which lead to the current state of the economy is the Reform and Opening period (改革开放 gǎigé kāifàng) beginning in 1978 and credited to Deng Xiaoping, who at the time was head of an important advisory committee in the Central Government.

Reform & Opening had a few key points that marked a significant change from the previous economy. One of the more significant of which was de-collectivisation of farming, which had been a major issue in the previous decades. The country also opened up to foreign investment at this time. It was in this period that the PRC went from the state it was in in the 1970s with a poor economy and little growth to a major economic power in areas like manufacturing. Must of the more "superpower" style development is more recent, so we can't really get into it here, but it was based on the growth which was afforded by Reform & Opening. Deng was not without his detractors, and it was also Deng Xiaopinng who ordered the troops to fire on protestors in 1989 in Tian'anmen Square, but today he is still widely regarded as the mastermind of China's rapid growth after the Cultural Revolution and Great Leap Forward. In effect, his policies undid much of the economic damage done under the earlier periods.

One thing to remember though is that this did not result in the decentralisation of many of the major industries. Much of China's economy in this period was still state controlled. While some of the major factories and industries did have points of privatisation, it wasn't the norm across the board.

There's another conversation to be had about what real communism is and if China ever really fit that, but that's better suited for /r/philosophy I'd imagine. The relevant point here is that China's economy was still ultimately beholden to the desires of the Central Government.

As an illustrative example, then as it is today there was no formal notion of private land ownership. You are simply leasing the land you "own" from the State for 70-year periods. Where that goes in the near future where those leases are for the first time set to expire is still to be seen. The main thing though is that, despite being Communist since it has the word Communist in both the country and party name, there are also a lot of remnants of communism that were retained into and beyond Reform & Opening.

hi OP. Just a note that this subreddit does not permit the discussion of current affairs, so responses will have to cut off at 1995, per the sub's "20-year rule". If you're looking for an analysis that includes more recent events/conditions, do consider x-posting this elsewhere, e.g. /r/AskSocialScience, an economics or political sub, or a regional sub.

How did China go from being a communist country to having the mildly intervensionist economic policy that it appears to have now?

You should keep in mind China does -not- have a "mildly" interventionist economy but a very statist one. The state owned companies contribute around 40% of the non-agricultural GDP in the entire country. When you start including enterprises of ambiguous ownership then it rises to over 50%.