Residency by Investment is when an investor who wishes to become a resident of a specific country does so by virtue of a financial investment, normally in the form of a real estate purchase or government bonds. Often an investor will initially have to become a Temporary Resident, with a limit on the amount of time they can spend in that country. After 3-5 years (depending on the country) an investor can apply for Permanent Residency which entitles them to remain in that country without limitation, be eligible for state benefits and be able to establish a business there.
After a year of permanent residency, an investor can then apply for full citizenship of that country. Only as a citizen will they become a passport holder, and benefit accordingly from the visa-free travel that a new passport brings. As a resident, one can benefit from that country domestically, but as a citizen, those benefits become international.