Diversity is making use of the fact that if you have two or more antennas positioned such that there is no or little correlation in the signals received from the same transmitter you actually can get some gain compared to the gain of a single antenna. It works for both static links and even better if the transmitter or receiver is moving.
Minimizing the correlation of the received signal can be achieved by space diversity, both in the horizontal or the vertical plane or by using polarization diversity.
* Space diversity is used for example in Private Mobile Radio, like MPT1327, DMR, Tetra, WiFI routers, etc but also in LTE where it's known as Mimo. (MiMo in LTE takes the concept of no or little correlation between two antenna signals quite a bit further). It also works perfectly with 2 or more omni directional antennas.
* Polarization diversity is used for example in GSM, UMTS etc. In a panel Antenna you put for one frequency band two antenna's. One at an orientation of +45 degrees, the other at -45 degrees.
Now where does the gain come from? Antennas seldom receive only the direct signal from the transmitter. In real life there are always objects reflecting the signal. The reflected signal(s) has/have a longer path length between Rx and Tx and will be often in a different phase then the line of sight signal. At the Rx-antenna the signals add up with their amplitude and phase.So it can happen that you end up with a reduced amplitude or no signal, an effect called fading. Now if this happens the chances that exactly the same happens at your second or third antenna are pretty slim.
With polarization diversity you make use of the fact that reflected signals often have their polarization changed as well.
So if you build multiple Rx-front ends fed by multiple antennas you can get some gain.
With horizontal diversity and omni directional antenna's separated > ~ 10 lambda you can get ~2,5 dB gain. Adding a third Antenna will get you roughly another dB.
Another reason to use multiple Antennas is reliability. Lightning can take out an Antenna. Chances that it takes out two Antennas at the same time are pretty slim. The same reasoning yields for antenna circuits deterioration due to water ingress in cables, loose connectors etc.
In LTE MiMo needs reflections to boost the throughput. Hence it does not bring much in a rural environment with less and weaker reflections. It's definitely worth the effort in a dense urban environment with lots of reflections.