The general idea is that make supports (reasonably) minimal rebuilds — i.e., you tell it what parts of your program depend on what other parts. When you update some part of the program, it only rebuilds the parts that depend on that. While you could do this with a shell script, it would be a lot more work (explicitly checking the last-modified dates on all the files, etc.) The only obvious alternative with a shell script is to rebuild everything every time. For tiny projects this is a perfectly reasonable approach, but for a big project a complete rebuild could easily take an hour or more — using make, you might easily accomplish the same thing in a minute or two…
I should probably also add that there are quite a few alternatives to make that have at least broadly similar capabilities. Especially in cases where only a few files in a large project are being rebuilt, some of them (e.g., Ninja) are often considerably faster than make.