-(으)시겠어요 is quite a polite form to make suggestions or to ask after his preference. It is generally appropriate to use this form whenever you want to make a suggestion or ask after someone’s preference when politeness and formality are important.
The form -(으)시겠어요 is a very formal and polite way to ask after someone’s preference or make a suggestion to someone. Perhaps you have noticed, but the form incorporates the honorific marker -(으)시-. Therefore, this form is reserved for situations where politeness and formality are important.
-(으)시겠어요 is still more formal than the honorific forms of -(으)ㄹ래요 and -(으)ㄹ까요, -(으)실래요 and -(으)실까요 respectively. Please bear this in mind when selecting which one to use. Depending on the situation one form is more suitable than another. Using a less appropriate form will be strange, but in the worst case, it might cause some frowning eyes. These things matter in Korea and its culture. You best get used to these rules.
가: 커피에 우유와 설탕을 넣으시겠어요?
나: 우유 말고 설탕만 주세요.
A: Would you like milk and sugar in your coffee?
B: Just sugar, please, not milk.
How To Use
The form -(으)시겠어요 comes with all the rules you have learned on how to use the honorific marker, -(으)시-. A quick reminder. The form follows the ‘batchim/no batchim’ rule. Keep an eye out for the Korean irregular verbs and certainly when using a verb of the ㄹ irregular verbs. Those verbs can be a bit more surprising than you think with this verb ending. Though it shouldn’t be something you cannot handle by now.
Furthermore, because it incorporates, the honorific marker, you have to use honorific verbs instead of their regular counterparts, meaning instead of 먹다 (to eat), you have to use 드시다 (to eat).