Learn Hardcore Korean: chinguege sajineul boyeo jwoyo.

에게 is the dative particle that marks the indirect object or recipient of an action. In 친구에게, it means “to (a) friend.” You could also say 친구한테 in casual speech—both mean “to my friend.”

Why is 사진 marked with -을 in 사진을?

-을 is the object particle attached to nouns ending in a consonant. It indicates that 사진 is the direct object of the verb 보여 주다—the thing being shown.

Why is it 보여 줘요 instead of just 보여요?

보여 주다 is a compound verb meaning “to show (for someone).”

보이다 means “to be visible” or “to appear.”
보다 means “to see/look.”
You need 보여 주다 (show + give/do for someone) to express “show to someone.” Dropping 주다 changes the meaning.

Can 보여 줘요 be contracted to 보여줘요, and is there any difference?

Yes. In spoken and casual written Korean, 보여 줘요 is often written and pronounced as one word—보여줘요. There’s no change in meaning or formality; it’s just smoother in speech.