The bill in question invokes the RLA, but specifically forces management and labor to accept a deal agreed on by a technical majority of related labor groups (that notably represents less than half of affected workers.)

Because they're forced by law to "accept" the new contracts, any organization/strike actions would not be legally recognized as such and therefore not protected as such - hence, "illegal."

Of course Time is favorable to the capitalist interests but https://time.com/6238361/joe-biden-rail-strike-illegal/

The Railway Labor Act gives the POTUS the ultimate ability to force one side or the other to take concessions in order to avoid a strike. For instance, he could force the bosses to give sick days, or he could force the workers to work without sick days.

He chose the latter.