Telefonumun ekranını korumak için bir ekran koruyucu takmalıyım.

telefonumun = telefon (phone) + -um (1st person singular possessive “my”) + -un (genitive case “of”).
So telefonumun literally means “of my phone.”

We use ekranını because:
1) ekran (screen) is a definite, specific object here (“the screen”). In Turkish, definite direct objects take the accusative suffix -ı/-i/-u/-ü.
2) That same suffix also shows that it belongs to telefonumun (“of my phone”).
So ekranını = ekran + -ı (accusative for definite object) and implicitly ties into the possessive genitive on telefonumun.

korumak fürn = koru- (root “protect”) + -mak (infinitive “to…”) + için (“for/in order to”).
Put together, korumak için means “in order to protect.”

bir is the indefinite article equivalent to English “a” or “one.”
By saying bir ekran koruyucu, you mean “a screen protector,” not “the screen protector.”

It’s a noun-noun compound: ekran (“screen”) + koruyucu (“protector”).
Literally “screen protector.”

takmalıyım = tak (root “attach/wear/install”) + -malı (necessity suffix “must/should”) + -yım (1st person singular “I”).
So takmalıyım means “I should/must attach (it).”

takmalıyım is the formal “I should / I must” using the -malı/-meli necessity mood.
A more colloquial alternative is takmam lazım, where lazım (“necessary”) follows the negated verb form: tak-ma-m lazım (“it is necessary that I attach…”). Both mean “I should attach / install.”