Learn Hardcore Turkish: Limonlu su beni yazın hemen serinletiyor. - Lemon water cools me down immediately in summer.

limon-lu su: lemon-with water → “water with lemon”
ben-i: I-ACC → “me” (direct object)
yaz-ın: summer-ADV → “in summer”
hemen: “immediately/right away”
serinle-t-iyor: cool(become)-CAUS-PROG.3SG → “(it) is cooling (someone) down” (or, in context, habitual)

Because beni is the accusative (direct object) form of ben (I). The verb serinletmek is transitive, so its object must take the accusative case: ben → beni. In other words, “It cools down me” → “It cools me down,” so Turkish marks “me” as beni.

Could I use bana instead of beni?

No. Bana is dative (“to me/for me”) and is used with verbs that take an indirect object (e.g., Bana limonlu su ver = “Give me lemon water”). Serinletmek takes a direct object, so you need the accusative: beni.

What exactly does serinletiyor mean and how is it formed?

Base adjective: serin = cool
Inchoative verb: serinle- = to become cool
Causative: serinlet- = to make (someone/something) become cool
Tense/person: serinlet-iyor = is cooling (3rd person singular) So serinletiyor means “(it) is cooling (someone/something) down; (it) makes (someone) feel cool.”

Why is the tense -iyor (present continuous) and not the aorist -ir?