You'll find here that in terms of European option prices, the absence of calendar arbitrage writes
$$ \frac{\tilde{C}(k\, F(0,t_2),t_2)}{F(0,t_2)} \geq \frac{\tilde{C}(k \, F(0,t_1),t_1)}{F(0,t_1)}, \forall k \in \Bbb{R}, \forall \, 0 < t_1 < t_2 \tag{1} $$
where $\tilde{C}(K,t)$ denotes the undiscounted European call price for strike $K$ and time to maturity $t$ and $F(0,t)$ the underlying forward price for delivery at $t$ as seen of $0$.
Suppose you would like to translate this inequality in terms of implied volatility i.e. by working in a Black-Scholes world. In that setting it is well known that
$$ \frac{\tilde{C}(k \, F(0,t),t)}{F(0,t)} =: \mathcal{C}(k,w) = N(d_+(k,w)) - k N(d_-(k,w)) $$
with
$$ d_{\pm}(k,w) = -\frac{\ln(k)}{\sqrt{w}} \pm \frac{1}{2}\sqrt{w} $$
where we have let $w = w(k,t) = \sigma^2(k,t) t$.
Then inequality $(1)$ can be rewritten as
$$ \mathcal{C}(k, w(k,t_2)) \geq \mathcal{C}(k, w(k,t_1)) \tag{2}, \, \forall k \in \Bbb{R}, \forall 0 < t_1 < t_2 $$
which is verified iff $\forall k \in \Bbb{R}$
$$ \frac{\partial \mathcal{C}}{\partial t}(k,w(k,t)) \geq 0, \,\, \forall t\in \Bbb{R}^+ $$
So that this translates to
$$ \frac{\partial \mathcal{C}}{\partial w}(k, w(k,t)) \frac{\partial w}{\partial t}(k,t) \geq 0 $$
where the first term is positive (see link with BS vega) hence the conclusion.