Sputnik was only launched a few hundred kilometers above Earth, and transmitted only a simple beeping signal. What was the purpose of having four antennas? Wouldn't one be powerful enough?
Sputnik was the first satellite. It was set up before we had an understanding of how difficult it would be to maintain a satellite's position, and in fact was a very simple system overall. The 4 antennas gives an omnidirectional broadcast pattern.
The simplest form of an antenna, a dipole, has two beams going in opposite directions. It's beam pattern looks like this:
If the direction wasn't pointed at Earth, it could end up in a null, meaning that the signal couldn't be received. The pattern that they gave allowed for Sputnik to be in any orientation and be able to transmit signal to anywhere visible on the ground below.
I think @PearsonArtPhoto 's answer misses several major points.
The satellite carried two antennas designed by the Antenna Laboratory of OKB-1 led by M.V.Krayushkin. Each antenna was made up of two whip-like parts: 2.4 and 2.9 meters (7.9 and 9.5 ft) in length, and had an almost spherical radiation pattern, so that the satellite beeps were transmitted with equal power in all directions; making reception of the transmitted signal independent of the satellite's rotation. The whip-like pairs of antennas resembled four long "whiskers" pointing to one side, at equal 35 degrees angles with the longitudinal axis of the satellite.
There were two transmitters, at 20 MHz and 40 MHz. Each transmitter was connected to only one pair of antennas and this is important. The opening angle between the two radiators in a given pair is 70°, close to right angles! This means that in the far field the polarizations of the two are nearly perpendicular.
The significance here is that if two waves have orthogonal polarizations, they would not strongly interfere with each other. You can (roughly!) add their radiation patterns algebraically - one antenna's null would be "filled in" to some extent by the power from the other antenna. If they were not close perpendicular, they'd have to be added strictly as vectors and that would just produce new and different null patterns.
One pair was connected to the 20 MHz transmitter, the other pair to the 40 MHz transmitter. These have wavelengths of 15 and 7.5 meters, respectively, so the radiation patterns would have been substantially different for each individual element of a pair, at each frequency. While the pattern wouldn't actually be spherical, the almost-perpendicular orientation of the two radiators in a given pair ensures that the null of one would tend to be filled in by the other, and interference was minimized thereby avoiding the creation of new nulls via interference.
A similar design was used by several NASA Explorer satellites as well:
above: Image of Explorer XVII from NASA
above: Image from here - original artwork on the circa-1958 plastic model of a Vanguard series satellite - antenna pattern is more easily visualize in this representation. More about Vanguard models can be found in Space Technology from Six Decades Ago: 1:5 Scale Hawk Project Vanguard Satellite.
above: Vanguard 1 NASA image from here. Three true dipoles appear to be mutually perpendicular.