It's spelled out explicitly in your OP. I already cited the passage. Is there some other conclusion we should draw, other than literally "Approving the new gun legislation was crucial to maintaining warm relations with the EU and a “No” could threaten that, the Swiss government warned before the vote."?
Ok. I will give you the lecture but don't complain about being patronized.
There are no border controls for people between the EU and Switzerland.
Think about what a border is. It is the line between two countries. Each of these countries can make its own laws without regard to the other. That's called sovereignty.
It may be legal for a person to possess a weapon in one country but not in the other. The same logic applies to all sorts of things like narcotics, hazardous materials. Then there are laws about taxes and standards and so on.
Think about this. Maybe one on one side of the border there is weak protection for intellectual property like brand names. Then one needs to intercept the counterfeit merchandise at the border or the brand owner will lose out on license fees.
Or maybe there is less drug enforcement. So drugs are interecepted at the border.
If a country wants to enforce its own laws on its territory then it needs to check whatever comes into its territory for compliance.
The EU and Switzerland have aligned their laws to the point where there was little to do at the border. So everyone could live with giving up border controls.
Unfortunately there has been some gun misuse which caused the EU to implement stricter gun control. Now the open border becomes an issue again. Its more liberal gun laws undermine EU sovereignty.
The obvious solution is to bring back border controls. Would an informed person call that "economic bullying"? Hmm?
As the swiss have now chosen to better control guns, that will no longer be necessary.
The border problem played a large role in the referendum. But there are many in Switzerland who favor stricter controls regardless. Eventually we don't know what the result would have been if the law had been debated on its own merits.