How does the engine wash improve engine performance?
A performance increase after a compressor wash is expected to happen, but it does not return to the original performance as there are more degradation mechanisms at play. This answer explains that organic build-up causes a change in airfoil shape causing the recoverable performance degradation. Another source may be deposition of salts and dissolved constituents to deposit on the blades as a result of ingesting salt water from which the water evaporates in the hotter stages of the compressor. E.g. erosion is a non-recoverable degradation (for a compressor wash). This is clearly shown in the figure below.



From: Modern Gas Turbine Systems, High Efficiency, Low Emission, Fuel Flexible Power Generation, Woodhead Publishing Series in Energy


  I wonder if the engine performance would improve significantly after having a wash.


The performance increase is considered significant enough, else we would not go through all the trouble of performing the procedure. Fuel costs is one of the major direct operating costs, reducing fuel costs by washing the engine doesn't only save money, it also reduces the amount of used fuel (nice green image for the airline).


  As it is said that the water should be injected at 1 o'clock position on the LPC (aft view), why does it need to be at that specific position?


This could be done to prevent build-up of water in the lower section; now it has to travel half the engine down, by the time it reaches that location the water will be transported aft by the churning blades. The water is therefore better distributed, which is confirmed by this practical answer.