What does ขอร้อง หัวใจ ลบภาพเธอที่มีให้หายออกไป
ก็รู้ จะลบภาพเธอที่มีให้หายอย่างไร mean?

ขอร้อง beg
หัวใจ heart, mind. It is more like mind here; we also have another word for mind จิตใจ.
ลบ erase delete
ภาพ image
เธอ her
ที่ is a preposition here. that
มี have
ให้ Normally, it is used as a verb; it means "give", but it is more like an adverb here. It emphasizes ลบ to give the imperative feeling and "... so that ..." meaning. For example, we could hear ลบให้หมด from a teacher telling students to erase something entirely, ทำให้เสร็จ from a boss telling his employee to finish their work.
หาย, หายไป disappear, to be lost
ออกไป depart, leave
หายออกไป is a phrase that Thais would not use but it could convey the "fade away" meaning
ก็ has no particular meaning and used together with other words to form different meanings. It is something like "I also know that" here with รู้
รู้ know. ฉัน (I) is not shown here. It is shortened from ฉันรู้
จะ Generally, จะ means "will", for example, I will do this ฉันจะทำอันนี้. But it should be translated as "can" here. จะ here is a shorten form of จะสามารถ
อย่างไร how

() represent the words that not shown in the lyric but Thais know that they exist to convey the correct meaning.

(I) beg (my own) mind. Delete her image that (I) have so that (it) fades away (from my mind).
(I) also know (that is what I should have done but) how can I delete her image that (I) have (from my mind)?

The last sentence implies that he still loves her.