In the first amendment, what is meant by “right to petition the government for a redress of grievances”?

Congress should make no law respecting an establishment of religion, or prohibiting the free exercise.

It means that the People, when wronged, are allowed to petition the government for a repair to the offending law, action, business, etc. 

Definitions (not relevant to the answer but may help with understanding)

Petition: A formal written request. It's almost always signed by a lot of people, all of whom support & advocate for the written cause.

Redress: To fix an issue, to correct it

Grievance: A wrong, be it real or not, that provides reason for complaint. 

Johnny johnny yes papa