Because they were the only colony that refused to hand over leadership to the blacks. Whites were only 5% or so of the population, but it was a way higher percentage than most of the African colonies. Basically they saw what had happened in all the other ones and said "fuck it, we're not seeing everything we built be destroyed". They declared independence, which was a massive embarrassment to the UK government.
This. The UK was outraged at what the Rhodesians called UDI, or Unilateral Declaration of Independence. However, since Rhodesia had their own army and air force, the UK could do nothing about it except push for UN economic sanctions. The sanctions were somewhat porous; Rhodesia traded a lot with South Africa, Portugal, Angola, and Mozambique at the beginning. South Africa provided military support/supplies, as well.
In 1974 there was a revolution in Portugal. The new government was tired of having large numbers of troops in Angola and Mozambique, so they left, granting both countries independence. The newly-independent Angola and Mozambique largely cut off relations with Rhodesia, allowing Rhodesian insurgents to use their countries as bases. In the end, the only external support came from South Africa, and even that support was tepid.
In the end, there was free elections, with Robert Mugabe elected president of Zimbabwe. Things went fairly well for a while, then went downhill fast. White farmers mostly left or were driven off their land, and the new black farmers produced little, not even enough to feed the country, much less export. There was famine or near-famine in Zimbabwe for a while, and the Zimbabwean dollar inflated to Weimar Germany proportions.
While Mugabe's replacement has encouraged white farmers to return, doubt many have returned. From what I know, the Zimbabwean dollar is still rather useless, with US dollars and other hard currencies used.