What does irasshaimase mean?

That means "Welcome" (normally used in shops/restuarants) in Japanese.

Irasshaimasu is an honorific verb (ie only used to people socially above you) that literally means "to be (in a place)". Thus if you want to ask whether someone is there respectfully you would ask "*** irasshaimasu ka".
The polite imperative form of the verb is Irasshaimase, and this is what is used in restaurants with the meaning of "Welcome", although the literal translation is "Be here".
The real feeling here is something like "Honour us by coming into our humble establishment".