Eating with the seasons is one of the oldest and least controversial pieces of dietary advice on this whole site — it predates refrigeration and long-distance transport, when there was, in a practical sense, no other way to eat. Ayurveda formalises the idea as ritucharya, seasonal routine, adjusting diet to the qualities of each season.
The broad Ayurvedic guidance runs: lighter, cooling, hydrating foods — fruit, cucumber, leafy greens — through the hottest months; warming, heavier, more grounding foods — root vegetables, whole grains, ghee — as the weather cools; and a gentler transitional diet at the changeover between seasons, which Ayurveda treats as the time the body is most vulnerable to illness.
Two separate claims are bundled together in "seasonal eating," and they hold up differently. That fruits and vegetables in season are fresher, more nutritious and cheaper than the same produce shipped from elsewhere or grown in artificial conditions is well supported and uncontroversial. The stronger traditional claim — that the body specifically needs different balances of warming or cooling foods as the seasons change, in the precise way Ayurveda describes — is a matter of traditional theory rather than established nutritional science. As practical advice goes, though, this is about as low-risk as it gets: eat what is genuinely in season locally, and you are very unlikely to go wrong.