The sputnik programme was introduced by Koroliov (previously doing rocketry) due to fears his team would "lose the priority" if the USA launched an artificial sattelite first (as they were planning to do for the international geophysical year).

source: S.P.Korolëv quoted on cosmoport.com.

Was it just the next logical step from missiles?

Exactly the reverse: Missiles are the next logical step from getting into orbit. The point of Sputnik was that, if you could get a satellite into orbit above the US, there was nothing stopping you from crashing it down into the US - and nothing stopping it from having a nuclear warhead. ICBMs are not prior to boosters that can get into orbit; the boosters are a prerequisite.

The Soviet leadership was actually rather nonplussed by the possibility of space exploration, at least initially. Korolev's team had gotten approval from Malenkov for development of an artificial satellite in 1954, but the regime's leadership was hardly enthusiastic about the possibilities offered by space exploration. When Korolev and others involved broached the topic to Khrushchev when the latter toured the R-7 rocket facilities in February 1956, the Soviet premier was not terribly impressed by the potential of using the R-7 ICBM for peaceful purposes. In short, the Soviet leadership primarily saw rockets as mostly as a military weapons. Korolev and other scientists involved in the development of Sputnik had to constantly lobby their superiors by emphasizing the role satellites could play in defense and that the development of satellites would not hinder the development of missiles for the strategic rocket forces.

The US's soft commitment to launching a satellite and the successful launch of a Jupiter-C rocket in September 1956 convinced Korolev that the Soviet's satellite program was behind the US. The result was that the Soviets scaled back their ambitious satellite, Object D, into something much more simpler and lighter, the PS-1. As before, the design team had to sell the program to the Soviet leadership emphasizing the American threat. Korolev's letter to the Special Committee noted

In September 1956, the U.S.A. attempted to launch a three-stage missile with a satellite from Patrick Base in the state of Florida which was kept secret, The Americans failed to launch the satellite ... and the payload flew about 3,000 miles or approximately 4,800 kilometers. This flight was then publicized in the press as a national record. They emphasized that U.S. rockets can fly higher and farther than all the rockets in the world, including Soviet rockets. From separate printed reports, it is known that the U.S. A. is preparing in the nearest months a new attempt to launch an artificial Earth satellite and is willing to pay any price to achieve this priority.

In reality, the US satellite program was pretty moribund and Korolev mistakenly believed the Jupiter-C launch was an attempt to launch a satellite. But the appeal that a successful US launch would publicly undercut the sense of Soviet military superiority and the simplified nature of the new satellite designs meant that Korolev's appeal for an accelerated satellite program was successful and the USSR Council of Ministers signed a decree in 15 February 1957 to launch a Soviet satellite.

The initial Soviet press reports on Sputnik's successful orbit were very low-key. The Soviet's news agency TASS's press release on PS-1 emphasized the boilerplate tropes that this development heralded the triumph of Soviet science without divulging much on the satellite's technical details. The article on the PS-1 in Pravda was not event the headline for the day's news.

What changed this lukewarm Soviet attitude towards spaceflight was the US's public panic over Sputnik and the global reaction to this advance in Soviet science. Western press reports quickly filtered back to the Soviet leadership and they finally realized the immense propaganda value that space exploration held for the wider Cold War. The Soviet press suddenly started a massive publicity campaign further explaining this triumph and Khrushchev invited Korolev into a personal meeting to discuss the future of the Soviet space program. Eight days after Sputnik's launch, Korolev received authorization to launch a satellite coinciding with the fortieth anniversary of the October Revolution the following month. Sputnik-2, containing a biological element, the ill-fated pooch Laika, launched on 2 November 1957. The rushed nature of the design contributed to the failure of the capsule to protect Laika and the nature of her death was a state secret until 1999. Yet, the propaganda coup reaped by space exploration could not be denied, and portraits of the Soviet space dogs became ubiquitous within the Soviet Union and Eastern bloc. Sending a man into orbit became the next logical step.

Source

Siddiqi, Asif A. Challenge to Apollo The Soviet Union and the Space Race, 1945-1974. Washington, D.C.: National Aeronautics and Space Administration, NASA History Div., Office of Policy and Plans, 2000.