In a very, very general sense, what Renaissance Technologies does well [and others try to do, many do less well] is understand where the "true" signal is (i.e. where prices should be) and what is noise (i.e. over-/under-reactions by others in the market) in the total signal of market prices. Generally, trading profits are made by taking the opposing position to someone who over- / under-reacts to where the market price is going to be because the market will "come back" to the true price.
Cryptographic algorithms and speech recognition algorithms have been developed to accomplish essentially the same thing ... it's necessary to separate noise from underlying signal in those and other applications of information theory, basically this is machine learning ... in general, becoming proficient in machine learning is skill that applies in many fields and for the mathematically-inclined individual, it is well worth studying [because there are so many applications, not just trading]. A good starting point would be the lectures by Andrew Ng of Stanford University
Attempting to beat the hedge funds really isn't something that should be copied by amateurs, i.e. it's an expensive way to get an education ... the algorithms are dynamic; the people using these algorithms must be faster, better, smarter than others who are using these algorithms ... continually updated, adjusted, advanced, accelerated, sharpened by teams of very smart people ... Renaissance isn't alone; it has smart competitors; there's a constant struggle by well-capitalized new firms with smart founders to make these adjustments with smarter, brighter, more capable teams of people who have access to better, faster technologies ... for example, many these algorithms must now execute so rapidly that it's necessary to use specifically-designed hardware that use ASIC integrated circuits or similar hardware, i.e. software on a supercomputer isn't fast enough to execute the trades ... the oscillation of the noise around the signal happens faster and faster.
Generally, the world benefits by the development of these algorithms and hardware, because the advances in this technology can eventually "spill over" into other projects in the rest of the world ... you aren't likely to find out what Renaissance Technologies and its competitors are doing primarily mostly because the people doing it don't have time [while they are doing it] to write papers and explain; it's not really that they want to keep secrets (i.e. there's some advantage to having your competition look at what you were doing last month, last year because by the time they understand it, you already know how to beat your old strategy), it's mostly that they don't have time / inclination to write up the explanation just yet ... as with all pursuits, the day eventually arrives where a person just doesn't really care any more and maybe wants to answer questions or tell the story.
Anyone who smugly tells you that they can't tell you because it's a secret basically is telling you they're just somebody like Mr. HR-clerical-job-app-screening-person and they couldn't begin to understand -- so it is easier for them to tell you that it's a secret. It's not that much of a secret ... mostly, it just happens fast ... so you need to be able to pick it up on your own without somebody explaining ... if you need to learn by having it explained, you shouldn't try to use it.
