A chosen name is a name someone picks for themself. This is usually so their name reflects them better as a person. This is different from a birth name or dead name. Transgender people generally pick a new name when socially transitioning because their birth name was gender exclusive or implied a gender they don't identify with. Other times their old name carries emotional baggage that they want to get rid of even if it was already gender neutral. The name chosen can be anything, like a gender bent version of their previous name, a gender neutral name, or a name that implies the gender they identify with.
Legal Name Change
Most jurisdictions allow individuals to legally change their name, usually by petitioning the court and appearing for a hearing. Through this process a chosen name can become a person's legal name as well. Documents like a driver's license, passport, birth certificate, or other government-issued ID get updated/re-issued to reflect this change.