From the New York Times, March 14, 1938:

While researching the annexation of Austria by Germany, I came across this speech given by British Prime Minister Neville Chamberlain to the Parliament the day after it happened.
In brief, Chamberlain says that on February 12, 1938, the Austrian and German Chancellors met and agreed on an extension of the framework set up by an earlier treaty. That treaty “provided, among other things, for the recognition of the independence of Austria by Germany and the recognition by Austria of the fact that she was a German State.”
The following week, Austrian Chancellor Schuschnigg “decided that the best way to put an end to the uncertainties of the internal situation in his country was to hold a plebiscite under which the people could decide the future of their country.”
On March 11, two Austrian Nazi members of the Austrian Parliament “presented an ultimatum to the Chancellor. They demanded the abandonment of the plebiscite and threatened that if this was refused, the Nazis would abstain from voting and could not be restrained from causing serious disturbances during the poll.”
“Later that day, feeling himself to be under threat of civil war and a possible military invasion, the Chancellor gave way to the two Ministers and agreed to cancel the plebiscite on condition that the tranquillity of the country was not disturbed by the Nazis.”
That’s how it was done. It’s probably why the far right and its leadership is getting away with so much in the US now.
Text of original speech:
FOREIGN AFFAIRS (AUSTRIA).
HC Deb 14 March 1938 vol 333 cc45-169
3.37 p.m.
The Prime Minister (Mr. Chamberlain) The main sequence of events of the last few days will be familiar to hon. Members, but no doubt the House will desire that I should make a statement on the subject. The result of the meeting at Berchtesgaden on 12th February between the German and Austrian Chancellors was stated by the former to be an extension of the framework of the July, 1936, Agreement. Hon. and right hon. Gentlemen will recollect that that Agreement provided, among other things, for the recognition of the independence of Austria by Germany and the recognition by Austria of the fact that she was a German State. Therefore, whatever the results of the Berchtesgaden meeting were, it is clear that the agreement reached was on the basis of the independence of Austria.
On Wednesday of last week Herr von Schuschnigg decided that the best way to put an end to the uncertainties of the internal situation in his country was to hold a plebiscite under which the people could decide the future of their country. Provision for that plebiscite is made in the Austrian Constitution of 1934. This decision on the part of the Austrian Chancellor was unwelcome to the German Government, as it was also unwelcome to the Austrian National Socialists themselves. Matters appear to have come to a head on the morning of 11th March when Herr von Seyss-Inquart, who had been appointed Minister of the Interior as a result of the Berchtesgaden meeting, together with his colleague Dr. Glaise-Horstenau presented an ultimatum to the Chancellor. They demanded the abandonment of the plebiscite and threatened that if this was refused, the Nazis would abstain from voting and could not be restrained from causing serious disturbances during the poll. The two Ministers also demanded changes in the provincial Governments and other bodies. They required, so I am informed, an answer from the Chancellor, before 1 o’clock in the afternoon. The Chancellor declined to accept this ultimatum, but offered a compromise under which a second plebiscite should be held later, with regular voting lists. In the meantime, he said, he would be prepared to make it clear that voters might vote for his policy but against him personally, in order to prove that the plebiscite was not a personal question of his remaining in office. Later that day, feeling himself to be under threat of civil war and a possible military invasion, the Chancellor gave way to the two Ministers and agreed to cancel the plebiscite on condition that the tranquillity of the country was not disturbed by the Nazis.