OK, first of all, here's the text of the 1st Amendment so that we both can agree what we're talking about because I see nothing where it "specifically states it is to keep the rights of the people":
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
The rest of this you should learn when you get to 10th grade civics, but they might not teach this anymore so I'll give you a quick lesson free of charge. The Bill of Rights and constitution are not the only laws of the land. There are thousands of laws passed over time on the federal, state, county, and municipal levels. The constitution and bill of rights are the yardstick to determine if these thousands of laws are CONSTITUTIONAL.
Since the 1st Amendment includes "the right of the people peaceably to assemble" there have been dozens (hundreds perhaps) of laws PROTECTING people's rights to peaceably assemble and these laws have been deemed CONSTITUTIONAL. Yes, these laws most definitely apply to PRIVATE COMPANIES. You can't own an apartment complex that only rents to whites or excludes blacks, Hispanics, Jews, or LBGTQRS+. You can't own a company where you only hire _____. Again, there are literally thousands of similar examples, surely you can find your own.
Get it? No charge for the civics lesson.