By itself, oku means "to put," but after a verb in the Te Form it means "will certainly do (that verb)," or "will go ahead and do (that verb)." There isn't a whole lot of difference between shite oku and plain old suru to express "will do," but shite oku, or any verb in the Te Form with oku, expresses the fact that someone will definitely do that something right away or in the very near future. It has a "will go ahead and do" kind of feeling to it.
Ron ni denwa shite oku. (I'll call Ron.)
Mado o akete oku. (I'll open the window.)
Kasa o katte okimasu. (I'm going to buy an umbrella.)
Kanojo ni ki o tsukeru you ni itte okimasu. (I'll tell her to be careful.)
Shukudai o shite okimashita. (I [went ahead and] did my homework.)