愛して is the te-form of the verb “愛する” meaning “to love”.
くれて is the te-form of くれる which can be used by itself as a verb to mean “to give me”. When it follows a verb te-form, it means “to do something for me” and the “something” is usually beneficial or favourable for the speaker so it implies the speaker’s appreciation for the action done by another person. So 愛してくれる means “to love me”.
ありがとう means, as you know, “thank you” and a verb te-form followed by くれてありがとう means “thank you for doing something for me”. Since くれて implies the speaker’s appreciation, It’s always attached to the verb when you appreciate someone for doing something. Many learners tend to miss くれて but you will never say 愛してありがとう without くれて.