Translation:I will look at that very closely.
The original German sentence does not imply "examination of the details" in my opinion. If it had been "Ich werde mir die [Einzelheiten/Details] [ganz/] [genau/] anschauen", yes. In general, if a source sentence contains a verb only and no noun, the best translation would be using those as well.
"sich anschauen" is a reflexiv verb, and a compound verb. E.g. "ich schaue mich an" = I look at myself, or "sie schaut sich an" = she looks at herself.
"anschauen" is a compound verb, meaning that it was compounded and gets sometimes separated when it is conjugated. E.g. "Ich schaue ihn an" = I look at him, or "Ich habe ihn angeschaut" = I looked at him.
"schauen" is a verb that describes a more general form of looking (not looking AT something). E.g. "Ich schaue Fernsehen" = I watch TV.
What is wrong with "I will watch that very closely"? The verb "watch" is given as an alternative but is rejected in the translation.