In public Thatcher claimed that Pinochet was unjustly accused, had been ‘kidnapped’ to Spain where he could not receive a fair trial, and in removing a ‘communist’ government had instituted ‘democratic order’, views all made clear in her Blackpool speech objecting to Pinochet’s arrest (nearly a decade after she left Downing Street), the full text published by the Guardian here. In many ways, Pinochet’s views were at least economically aligned with her own, as both were broadly neoliberals in the sense that they prioritised shifting to privatisation and the free market over government-funded services.

For some background, Britain had long supplied arms to Chile (including to the socialist Allende government, and the Hawker Hunter jets used during the coup against him that placed Pinochet in power), but the Labour government had placed arms sanctions on the Pinochet government in 1976. Pinochet was a neoliberal who followed the works of Milton Friedman and the Chicago school of economics, installing Chicago graduate Sergio de Castro as minister of finance (Ministro de Hacienda) in 1977, as well as others to high positions, who were known (along with others in Latin America) as the ‘Chicago Boys’. Thatcher was highly influenced by the same school - she considered Friedman to have revived the ‘economics of liberty’ and even called him an ‘old friend’, and in turn she saw Pinochet as an economic liberator of Chile. In this context, Thatcher relieved the arms embargo and allowed arms trading with Chile to resume in 1980.

But one point she dwells on in the above speech is, of course, the fact that Pinochet helped the UK during the Falklands War. In 1982 the Chilean government officially refused to endorse Argentina’s claim on the Falklands/Malvinas (the only Latin American country to do so apart from Colombia), and even allowed the British SAS to pass through Chilean territory to conduct operations in the war itself. Part of the reason for this support, beyond the improved relations with the U.K. due to economic policy, was that Chile was in turn in a dispute with Argentina over the Picton, Lennox and Nueva Islands (which had been awarded to Chile by the British government’s arbitration in 1971), the so-called ‘Beagle Conflict’ (after the channel by which they are situated). This had led to a quasi-conflict dubbed Operation Soberanía in 1978, where the Argentine navy moved to occupy the islands, though they aborted within hours. So Chile and the U.K. both saw Argentina as a threat to their territorial claims in the region.

It was only after the fall of the Argentine junta (officially the ‘National Reorganisation Process’) in 1983 that the Pinochet government signed a ‘Treaty of Peace and Friendship’ in 1984. Throughout the 1980s and beyond (they both left office in 1990), Thatcher and Pinochet remained on warm terms. In fact, later in life, after his ouster, Pinochet sought refuge in London and despite his arrest there in 1998, and an attempted extradition to Spain, he was released in 2000. He and Thatcher remained friends, and she even gave him gifts, including a silver plate and scotch.

As for ‘how did that work’... there’s an official answer (without reading her mind, this would be how she defended her stance in the speech above), and a practical answer. Pinochet’s abuses and many of the killings his regime and associated ‘Caravan of Death’ conducted or approved are well documented, and he also came to power as head of a military junta, so that Thatcher’s criticism of the Argentine government as a military junta could be seen as hypocritical, eg from her speech to Parliament just after the Argentine landings:

These discussions are complex, changing and difficult, the more so because they are taking place between a military junta and a democratic Government of a free people—one which is not prepared to compromise that democracy and that liberty which the British Falkland Islanders regard as their birthright.

Though in at least some official cases it is worth noting it was muted (e.g., welcoming the establishment of a more representative Argentine government but not wishing to interfere in the affairs of another country, as in her answer to a parliamentary question some months after the war from the official archives here.)

But I’m not sure what sort of explanation would ever suffice completely, so down to the practical answer: for reasons of national self-interest, it’s quite normal and common for government stances to be arguably hypocritical where parties may or may not help them. Rightly or wrongly, similar accusations have been levelled for decades at many governments worldwide over alliances from those during the Cold War to those due to economic ties with China or Saudi Arabia. One can even point to the fact that the Soviet state was a dictatorship under Stalin that invaded Poland shortly after Germany did, and killed millions of its own people, but was later treated as an ally by the UK and US when Germany invaded. For much the same reason, democratic Finland chose an alliance with Nazi Germany because they saw the Soviet Union as the greater threat to their own country. This was all, simply put, realpolitik: rightly or wrongly, and hypocritically or not, a government is of course more likely to criticise a dictatorship or junta with whom they are at war for territorial reasons, than one that supports them or whose trade they depend on.

A reference for Chilean support of the UK during the Falklands War: Freedman, L. (2004). The Official History of the Falklands Campaign.