Learn Russian: Летом он надевает шляпу.

Лето is the noun for 'summer', but to say that something happens in a season, Russian uses the Instrumental case without a preposition. So, лето changes to летом to mean 'in summer'. The same rule applies to other seasons, like зима (winter) becoming зимой (in winter).

Why use надевает here instead of одевает? Don't both mean 'to put on' or 'to dress'?

This is a classic tricky pair! Use надевать (to put on) when you are putting an item of clothing onto a body. Use одевать (to dress) when you are dressing a person (like a child or a patient). Since he is putting on an inanimate object (a hat), надевает is the correct verb.

Why does the word for hat end in '-у' (шляпу) here?

The dictionary form of the word is шляпа, which is a feminine noun ending in -а. Because the hat is the direct object of the verb (it is the thing being put on), it must take the Accusative case. In the Accusative case, feminine nouns ending in -а change their ending to -у.

Could I put летом at the end of the sentence instead of the beginning?

Yes, Он надевает шляпу летом is grammatically correct and means the same thing. However, putting time or location words at the very beginning of a sentence to 'set the scene' is extremely natural and common in Russian word order.