What gives the police the right to raid your home?

Probable cause is a reasonable belief that criminal activity is afoot.  It can be based on hearsay; for example, from a reliable informant.  A sufficient showing of probable cause to a judge results in a search warrant being issued to search a home.

A search warrant is not required if the police are in hot pursuit (chasing a suspect who enters the home); incident to an arrest; exigent circumstances (emergency situation where the time to obtain a search warrant may endanger the police or public), auto search (where the contraband and auto may be gone by the time the police return with a warrant), etc.

If the police are searching a home and discover other contraband in plain view, that is admissible evidence even if it wasn't the subject of the search warrant or the search was without a warrant for one or more of the reasons mentioned above.

Probable cause can be established without the police being inside a home and observing a law being broken.

Unless there is--

1) Some reason to think that someone is in mmediate danger (e.g. a cry for help or screams)

2) A criminal seen/chased by the police into the home

3) A visible crime (e.g. as police walk by on the sidewalk, glancing up, they can see someone freebasing cocaine or brewing up crystal meth through the living room window)

4) Permission from some other resident (e.g. a roommate) to enter

--the police would need a warrant. However, that warrant may be based on what you call "hearsay"--the testimony of other people, that they believe there is criminal activity or evidence of a crime--so long as it is credible enough to convince a judge that "probable cause"--that it's reasonably likely there's a crime--exists.

Related article: Can the police come and arrest me in my home?