进来 is come in.
请进 is please enter
where is the verb!
进 by itself is the verb enter. 进来 means "come in" and the speaker is already in the room; 进去 means "go in" and the speaker is outside the room.
进is the verb for "enter". 请 is the polite request word, the "please" here.
进 is the verb; it's just a shortened version of 进来 in this case. Both should be accepted here, but sometimes the meaning is subtly different. This is super common so expect to see more in future lessons.
Can this be used for other instances, such as welcoming friends or family into your home? Or is it only in the context of a restaurant or other public venue?