Let's say you are building your own app for some project and need a calendar feature. Writing hundreds of lines of code to calculate leap years and track days of the week is extremely annoying/tedious.
Thankfully, someone else has already perfectly solved that problem and put their code on the internet for free. A package manager is the tool that lets you easily grab that code. Think of it like an App Store specifically for developers. When programmers write useful code, they bundle it up into "packages" (or libraries). A package manager is a command-line tool that finds, downloads, and installs these pre-written chunks of code directly into your workspace.
If you write Python, you usually use a tool called pip. If you want to make a game, you can literally just type “pip install pygame” into your terminal, and boom, a ton of files/tools are added to your project to make building a game much easier. If you write JavaScript, you use npm (Node Package Manager). Typing npm install react pulls in everything you need to start building React pages.
The best part is the "manager" aspect. Often, the package you want to use actually relies on three other packages to work. And those three might rely on like ten more. Tracking down all those connected files by hand would be an absolute nightmare. Your package manager handles that messy part automatically. It figures out every single dependency your requested package needs, downloads the correct versions of all of them, and organizes them perfectly. Ultimately, it takes care of all the background file management so you never have to reinvent the wheel when starting a new project.