How can a Zener diode maintain a constant voltage?
Why does the voltage across it remains constant even when reverse bias is increased.?

It doesn't. The voltage increases very slightly as the current is increased.
To use a hydraulic analogy, it's like a weir across a river. While the water is below the crest of the weir, no water flows over. Once the water exceeds the crest, water flows. If you send a much greater flow than that down the river, the water height above the weir will increase only slightly.
The rate at which the voltage increases for increasing current is variously called the slope resistance, or the dynamic resistance, or the incremental resistance. This is usually listed at a particular current in zener diode data sheets, and is often in the ohm range. The 1N4278A 6.2 V diode has for instance typically 2 ohms slope resistance at 41 mA, and warranted less than 700 ohms at 1 mA.