If I buy (for example) frozen broccoli in the store with a hechsher, is it required to check for bugs? Does the hechsher signify that the vegetables have already been checked for bugs and are free of them? Or does it signify that the vegetable is kosher as long as I do not find any bugs during my bedika?
This depends on the Hashgacha. Some Hashgachot on broccoli are only signing off on the purposefully present ingredients and processing equipment, but are not addressing the requirement for Bedika (Triangle-K for example on a lot of frozen produce (See here)). Others are signing off that they indeed already performed a Bedika (typically via Chazaka) and that one is free to use immediately without performing an additional Bedika. (The Hashagachot appearing on Bodeik and Golden Flo brand frozen broccoli - as indicated by their packaging). These same Hashgachot will also certify frozen vegetables which do not require a Bedika (ex: frozen peas, carrots, corn). For these types of vegetables there really isn't any difference between the role of the various Hashgachot. Most Hashgachot will gladly share the standards behind their symbol if you contact them.