Since the beginning of the UFC, though not necessarily MMA, BJJ has proven itself as an effective grappling style within Mixed Martial Arts. This is not to say that styles such as Sambo, Judo, and Wrestling have not been proven effective as grappling styles in MMA.
Sambo, Judo and Wrestling all engage opponents with a different mindset and strategy. Sambo is a very effective mix of Judo (as its founder was a student of Jigoro Kano) and Wrestling. The explosive movements allow Sambo to be useful in a combat setting. Sambo though has the severe limitation of not allowing chokes within their competitive setting. Judo utilizes explosive level changes, throws, slams, chokes, submissions, sweeps, pins, and groundwork. Judo (the parent style of BJJ) was previously quite effective; but the time limitations on the ground (often no more than 30 seconds) as well as the newer 2011 rule set (IJF competition, not Traditional Kodokan Judo or Freestyle Judo) of not allowing hand-leg takedowns, and not allowing leg locks has limited it for use within MMA. Wrestling has an effective explosive stand up, ground, pressure grappling, allows for throws from a ground position, no GI and pins. Wrestling though suffers from a number of limitations including the lack of submissions (Roman-Greco), chokes, and the lack of effective techniques fighting from the back.
BJJ is not without its limitations such as not being as effective with throws when compared to Judo, Sambo, and Wrestling. BJJ rule set of making slams illegal and from a pin are a benefit and a detriment. It keeps fighters around longer, but makes them particularly ineffective going against superior throwing grapplers (which might injure them from these throws). Often through the throws and takedowns done by BJJ fighters will beat many other fighters. Within BJJ fighting from the back is typically what puts it on the forefront of grappling within MMA. It is here, where BJJ succeeds far beyond that of other grappling styles and makes it a consistent leader within MMA. BJJ consistently teaches its practitioners how to effectively fight from typically poor positions (side control, north south and the mount). The use of the various guard techniques makes BJJ effective going against a stronger opponent. The consistently use and focus of some fighters to utilize technique over strength (which is great if a fighter is tired). The utilization of inductive rather than deductive teaching methods (which leads to rapid development) is an advantage over other grappling arts.
The last reason is training and availability. Judo is a popular martial art would wide, but the throws and takedowns sadly take many fighters out of the martial art (the wear and tear on the body can be traumatic). Competitions are available but the amount of competitors is not as big locally and regionally. Sambo also suffers from the lack of available schools and competitions outside of the Russia and satellite nations. Wrestling is also popular worldwide but suffers from a severe limitation of training centers and schools outside of a scholastic setting (once you graduate from school where can you compete or train recreationally). An additional problem is the lack of competitions outside of a scholastic setting. BJJ also has a distinct advantage of having more available schools and competitions than all the other styles listed. BJJ is growing exponentially, many grapplers cross train into BJJ including high level wrestlers, and Judoka (Travis Stevens, Garry St. Leger, etc.). Some competitions even allow fighters to fight professionally (for money) and make a living from their training.
All of these reasons make BJJ, the best base martial art for Mixed Martial Arts. This does not mean that you shouldn’t cross train in other grappling arts. In fact, you should utilize BJJ with another grappling art. So if MMA is your goal, find a school an effective school that will maximize your potential.