He had set his sights on buying the painting and refused to give up.

set 在这里是动词，意思是"确定、固定"。 his 是代词 he 的所有格形式，表示"他的"。 sights 是名词 sight 的复数形式，原意是"视线、视野"，这里引申为"目标"。 on 是介词，表示"关于、针对"。 set one's sights on 是一个固定短语，意思是"把...作为目标"或"决心要..."。