Why was the Prime Minister of Australia dismissed in 1975, and what was the reaction?

I don’t know how much background information you have so I will provide additional details. You should be aware that although what actually happened is broadly agreed, discussion of motivations and reactions is still politically partisan.

The position of Prime Minister in Australia is not an elected position. The Commonwealth of Australia has two houses of government, the House of Representatives and the Senate. Both are elected and all adult Australians are required to vote in both elections, but the elections are different, at different times, and can result in different political parties controlling the different Houses. To be passed, legislation is initiated in the House of Representatives, must be approved there, then pass to the Senate, be approved there, then be signed into law by the Governor General as the representative of the Queen.

After an election of the House of Representatives (the lower house) the leader of the political party, or the agreed leader of the coalition of political parties which control a majority (more than half) of the positions in the lower house and has an expectation of getting legislation through the Senate (the upper house) goes to the Governor General and says that s/he is able to form a government. The Governor General then approves him as (the Queen’s) Prime Minister for Australia. The Prime Minister then appoints elected Members of Parliament as Government Ministers by giving them control of the different government (public service) departments and selects an inner group of the more important Ministers as advisors (the Cabinet). So in American terms, the Prime Minister is a politician who has won election in his own electorate, been selected as leader (by internal party procedures) of the party or coalition which won the last election and consequently been appointed head of the Administration by the Governor General.

If the Prime Minister loses the leadership of his/her party or coalition (by internal party procedures) s/he loses the Prime Ministership and the new leader may be appointed as Prime Minister. If the government party or coalition loses the majority in the Lower House s/he loses the Prime Ministership. If the party controlling government is unable to get legislation passed by the Upper House, specifically by the Lower House submitting a Bill to the Senate three times and having it refused three times, so the Houses are deadlocked, there must be a Double Dissolution. That is, both houses must go to a full election of all members, enabling the voters to decide on a new government.

Just to make things more complicated; although the Governor General is appointed by the Queen, the appointment is on the recommendation of her Prime Minister. So effectively, the Governor General is appointed on the say so of the Prime Minister. At the time in question, the Governor General, Sir John Kerr was a long time Australian Labor Party supporter, appointed on the recommendation of Prime Minister Gough Whitlam.

Prior to the events of the Dismissal, the Labor government had been racked by both sexual (the Morosi Affair) and financial (the Khemlani Affair) scandals and were in a poor situation to win an election, but were not due to hold one and were unwilling to do so. The opposition coalition (the Liberal Party and Country Party alliance) obtained a majority in the Senate and refused to pass the Supply Bill which was necessary for the Budget and without which it would be illegal for the government to spend any money such as government salaries, social security, Defence etc.

Whitlam tried to prevent an automatic Double Dissolution by not submitting the Supply Bill to the Senate for the third time, and in the interim wanted to get the Governor General to declare a half Senate election which might break the deadlock in his favour. Whitlam believed that he controlled the Governor General and that he could get the Governor General replaced if he wasn’t sufficiently amenable. In the interim the Whitlam government was unable to govern.

The Governor General, Sir John Kerr, obtained agreement from the then Leader of the Opposition, Malcolm Fraser, that if he was appointed Prime Minister, he would immediately get the Supply Bill passed in the Senate and request a Double Dissolution from the Governor General. The Governor General then summoned Whitlam and told him he had been dismissed as Prime Minister.

The reaction in Australia was mixed and included; surprise that the Governor General had that much constitutional power (he does), Shock and anger by the Labor Party faithful that they had been stabbed in the back by their own man (typified by the Whitlam rant “Well may they say ‘God Save the Queen’ but nothing will save the Governor General"), relief by most Australians that government was back to business as usual, followed by, as r/Ratslinky mentions an overwhelming vote against the Labor Party at the Double Dissolution elections.

Note:

Constitutionally a Prime Minister who is unable to secure supply he is required to call a double dissolution election.

Is not quite accurate, but the effect is the same. He must secure Supply to govern. It is the three refusals by the Senate that require an automatic Double Dissolution and that didn't happen. In this case the Governor General called the Double Dissolution on the advice of the newly appointed Prime Minister.