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In this diagram of JFET transistor the depletion region is narrower at one end and wider at other end. Why is that so? Is it due to the voltages applied at gate and drain?

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Yes it is - the depletion will be wider at the drain because it is connected to the positive of the battery whereas the source is at a voltage closer to that on the gate. As the gate gets more negative the depletion region will grow and eventually the device will be deemed to be "cut-off" and exhibit only low leakage currents from drain to source. This is an N channel JFET. P channel JFETs operate the same but with both batteries reversed. Having said that some N channel JFETs will operate "about" the same when D and S are reversed in polarity - these are sometimes used as voltage controlled resistors.

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