Interlocking is usually mechanical or relay based. In such interlocking systems, the various routes and switches outside a station are physically connected together so that setting one route through the station automatically disables all other routes.
For example, in a station with 2 loop lines and a main line, if a train is going to pass through the main line without stopping, the main line signal will be set to green and the switches will be set so that the train stays on the main line. In a mechanical interlocking, the points and signals will be connected in such a way that turning the signal green automatically prevents the points from being changed until the signal is...
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changed to Red. This prevents accidental changing of points while the train is passing through. Relay based systems also do a similar thing where the electronic switches operating the points are connected in such a way that switching on a few switches automatically disconnects other switches.
In these systems, if the routes in the interlocking have to be changed, say, after construction of a new loop line, then all the mechanical/electrical connections have to be relaid again. In a Solid state interlocking, all the switches controlling the points and signals are controlled via software running on microprocessors ( Hence the name 'solid state', which is the general term used for semiconductor devices like microprocessors). In these systems, changing the routes can be done purely through software changes and does not require any physical modifications in the Yard.

It doesn't directly increase the MPS or Capacity. It is mostly used for more flexibility and in general, it is a more advanced form of interlocking.
The advantages are as follows:
1) Flexibility and construction time- As I mentioned earlier, Solid state interlocking provides a lot of flexibility. In a normal interlocking, every modification requires physical work to be done. Wires have to be relaid and sometimes the tracks themselves have to plucked up and relaid to incorporate any new loop lines or additional signals. For a Solid state interlocking, as everything is...
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done in software, altering the routes is just a matter of updating the software, which can be done in a few hours and doesn't need any physical work to be done on the location. This also reduces the time taken for making changes or commissioning a new interlocking system, as a physical system would require a long procedure involving tenders, contracts, construction etc.
2) It can indirectly increase capacity as well. Usually, in most stations, trains end up waiting at the outer even if platforms are available. This is because the interlocking system automatically blocks a few routes. There may be a few empty platforms available at the time, but because of the interlocking, some platforms may not be accessible as the routes would conflict with trains on other platforms. Now, to improve this situation, in a normal interlocking system, one has to remodel all the tracks in the yard to provide accessibility to the empty platforms. However, in a solid state interlocking, they can just implement the changes through a software upgrade. This can practically be implemented in a few hours and can reduce congestion in these stations.
Almost all high speed, high traffic networks in the world now use solid state interlocking.