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I read in the lecture slides to a VLSI course that clock gating also can lead to a decreased chip area size. In my understanding, clock gating decreases the chip size as this technique requires an additional FF + and gate.

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If there is no clock gating a feed-back MUX is used to keep the output the same if the enable is low. One for every FF.

With clock gating they can all be replaced with an AND gate.

Additionally clock gating means lower power which means a leaner, smaller power grid.

The diagram below is an illustration of the difference.

I don't know where your "additional FF" comes from.

schematic

simulate this circuit – Schematic created using CircuitLab

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