There's a lot left unspecified in this question, since it is stated without precision, but the effective idea of the answer given here is that those jumps introduce extra variation into the forward distribution of the underlying. And such variation is the bread-and-butter of option value.
That said, the ambiguity in the question leaves room for other interpretations. In particular if you as a market-maker sold an at-the-money call for $100, and them immediately after your sale everyone found out that the underlying had a 50-50 probability of jumping down by half tomorrow, you would be very happy, because the underlying would drop in value by 25% or so and the option would go far out of the money.
So, what the person who said the value increases meant was, given two ATM options on separate underlyings with the same continuous volatility, and where the second underlying also had some downward jumps, the latter option will have higher fair value.
Mathematically, this ends up being associated with the second underlying having higher risky drift.