The transistor was the revolutionary replacement of the vacuum tube, which had been at the heart of computers for the first half of the 20th century. Vacuum tubes themselves had only two main problems: They were power-hungry and they were relatively big. Relative to their replacements, that is. They also had a tendency to burn out or leak during operation, which could prove disastrous.
In 1947, John Bardeen and Walter Brattain, along with William Shockley, their boss at Bell Labs, successfully amplified an electric current, using germanium. This "point-contact transistor", as it was called, was soon used to speed up computing and to make computers smaller and more efficient.
A good example of the transistor and the vacuum tube is the construction of the Manchester Computers, developed at the University of Manchester. The first, the Small-Scale Experimental Machine (SSEM) (developed in 1947), was a state-of-the-art testbed for new innovations in computing, such as the Williams tube. But it still used vacuum tubes. It had 550 valves and took in 3500 watts of power.
The SSEM's successor, the Manchester Mark 1 (developed in 1949), was much more powerful. It used 4050 valves and consumed 25000 watts of power. Yet the aptly-named Transistor Computer, built in 1955, used only 200 transistors and 1,300 diodes, and only used 150 watts. It wasn't the first computer to exclusively use vacuum tubes, but it was a huge step forward.
It's tough to say exactly why the transistor was created when it was (I'm answering the last part of your question now), but it could be argued that the computational advances of World War II (such as the Harvard Mark I) ensured that many new advances would be made in computing; the transistor happened to be one of them.
The integrated circuit
The integrated circuit, developed about a decade after the transistor, also had profound effects on computing. It was developed by Jack Kilby in 1958 - though many others were involved along the way, and there are disputes as to who should get the credit for inventing it first - at Texas Instruments. He used semiconductors to create an entire computer chip - the integrated circuit.
An integrated circuit can contain incredible amounts of transistors, and it is this complexity and compactness that make it so useful. Manufacturing was also much easier and quicker. Integrated circuits started a second computing revolution, which laid the groundwork for cheaper computers that could be available to the masses.
Now that the question is focused on the transition between vacuum tubes and transistors, I'd like to add something about semiconductors, because they play a key role in transistors.
Semiconductors allow for good conduction of electricity, but one of their really useful properties is that their conduction can be modified in a process called semiconductor doping. This adds "impurities" to the semiconductor, adding either electrons or holes. Semiconductors can be either n-type or p-type - n-type semiconductors have an excess of electrons, while p-type semiconductors have an excess of holes. These can be combined to form a diode.
Another relevant development was the creation of the Czochralski process, which allows crystals to be grown for semiconductors. This also involves semiconductor doping, and has allowed for semiconductors to be produced on a large scale, making it easier to build transistors.
Are there other technologies that have been crucial to computer development? Of course. I could cite the vacuum tube, cathode ray tube, solid-state drives, and a host of others as crucial to computer development. But the transistor and the integrated circuit were the two main players in the development of the "modern computer" in the relevant time period here - the mid-20th century. I suppose you could make cases for other technologies, but I'd certainly rank them at the top of the list.