From what I recall (there's been plenty of videos of Linus Torvalds discussing the origins of Linux), Linus, while at University or college, was doing a lot of work using Minix ("Mini-Unix," another Unix-derived system, where Unix had been around since the 1970's). As a kind of pet project during this time, He began developing His own kernel in order to address some of the things He thought were flaws in Minix (e.g. the fact that Minix had a micro-kernel architecture - i.e. the core of the system was very small in size, and a lot of the functionality existed outside of the kernel in other bits of code).
Linus perceived a monolithic kernel architecture (i.e. all of the core functionality, tasking, network activity, memory management, etc. to be within a single, larger kernel) to be faster, more stable and more secure (check out the "Tanenbaum-Torvalds debate"), so developed Linux around this in the early 1990's. For all intents and purposes, technically, Linux refers explicitly to the kernel, though it is often used to refer to a Linux distribution, or whole operating system.
A lot of the functionality was based on Unix functionality, such as pipelines, etc. and He kept the open source ethos, too, which allowed Linux development to grow into a community effort. Linux can be described as being Unix-like, because of the usage of some of the ideas develop in that, in the same way that Apple's MacOS can be described as being Unix-like, and even early MS-DOS, as the precursor to Windows, can be described as Unix-like.