Ye-152A at rest at an airfield (Date and location unknown)

I have seen these early MiGs with the central, protruding inlet cones since I was a child and never really wondered why those cones are there as compared to the blunt nosed MiG 15

or the F-86 Sabre

Yesterday I came across this exact question on Quora  and it shed some light on why those nose cones are so prominent. The answer of course is that the MiG 15 and the F-86 were subsonic while the MiG 21, 23 and the above pictured Ye-152 are supersonic. 

The plane goes supersonic, but all conventional jet engines - except for scramjet - require subsonic air to operate. There is some serious slowing down of air required because supersonic air does not treat turbines nicely.

The inlet cone diverts the sonic shockwave away from the inlet. The tip of the inlet cone - also called the spike - creates a conical sonic shockwave. The jet pushes this shockwave in front of itself while flying supersonic. The spike is shaped to keep this supersonic boom away from the engine intake.

As the aircraft accelerates this shockwave gets squeezed narrower to the point where it threatens to bite the lip of the inlet. High mach airplanes like the SR-71 pull the spike backwards to keep the shockwave diverted from the inlet even at high mach speeds. 

Depending on speed the air entering the inlet is already subsonic or low supersonic. Behind the cone is a diffuser that allows the air to expand slowing it further down and bleeder valves to get rid of excess air if needed. 

Modern supersonic aircraft achieve the same effect with a series of adjustable (or fixed geometry) shock ramps. These basically divert air to generate turbulence that creates shock waves to make sure the supersonic to subsonic translation happens before the air hits the engine fans. 

Further reading: how supersonic inlets work.