-(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 is strange, but in the worst case, it might cause some frowning. These things matter in Korean.
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. However, a ㄹ batchim is treated as if it were a no batchim. 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).