"Let's eat at this restaurant."
Translation:이 식당에서 식사합시다.
Although both verbs are technically correct, 씩사하다 is more formal than 먹다. So 씩사하다 is a better selection for the speech level--formal polite (합쇼체). In English, the best equivalent that I can think of is "eat" versus "dine."
씩사 also means "meal" or "repast." In a way, you could think of 씩사하다 as "to have a meal," which also sounds more refined than "to eat."