The short answer is to protect your security and privacy. Anyone can use publicly available IP lookup tools to find both your ISP and your general physical location.
On the irritating side of things, your IP address provides an way for invasive advertisers to bombard you with localized ads and to tie any collected browsing data to your IP to help track all of your activities.
Some folks may have stronger motivations to hide their IP addresses. This includes people living under repressive regimes that could use an IP address to hunt down political opponents, people escaping from abusive relationships, and anyone else who fears they could be targeted by malicious individuals that would love to get their hands on a physical location.
Identity thieves would also love to get their hands on this info to further exploit whatever data they may have on their target.
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Even if you don't fall into these categories, determined hackers can exploit your IP address to access your home network and anything connected to it, especially if you haven't properly secured your equipment.
In addition to the safety and privacy concerns mentioned above, a new reason to hide your actual IP address has arisen in recent years: geographic streaming restrictions.
Many streaming service providers will limit which geographic markets can stream which content, due to licenses to stream shows in some countries but not others, and obstacles like sports broadcasting markets.
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Many people choose to circumvent these geographic restrictions by using a VPN (explained in more detail below) to show the streaming provider's systems an IP address from a compatible country or region.
Most decent VPNs provide several geographic locations to choose from. Selecting one in the correct locale broadens what can be streamed via services you're already subscribed to and helps avoid pesky broadcasting blackouts that are mostly holdovers from the days of antenna-based TV.