TL;DR: While Intel had their heads shoved up their ass making the Itanium architecture, AMD made a 64-bit variant of x86 that was backward compatible with the older x86 ISA. Technology moved on, and amd64 was adopted while Intel kept trying and failing to push their binary-incompatible architecture.

Eventually, Intel had to give up and adopt AMD's amd64 ISA. In exchange for letting them use it, Intel lets AMD use the older x86 ISA.

VIA also built x86 CPUs for some time, they have a license as well; the issue with modern x86_64 is though that basically, you need licenses from both AMD and Intel. They do have a cross-license agreement, but there's no single point of contact for all licenses for a modern x86 CPU.

To sell into the US government in the 70s you had to ensure your parts could be sourced from a second company. That way, if you had supply problems, the government would just got to the second source.

AMD was a second source for the 8086.

So way back in what? the late 70's or early 80's, IBM decided they wanted to get into the microcomputer business. They didn't want to throw a lot of money at it developing it in-house, so they slapped together a machine from off-the-shelf components to include an Intel CPU, after failing to get the attention of the guy who wrote CP/M they hired some little nobody software house called Microsoft to do the operating system which they licensed on a non-exclusive basis, and figured the copyright on their firmware (the BIOS) would keep the system proprietary. It didn't. Compaq created and sold a compatible but non-infringing BIOS, which meant IBM had no legal standing to prevent anyone from building or selling machines 100% compatible with their line of PCs.

IBM had accidentally created an open standard, which wasn't so good for IBM but great for customers. You could price shop. There was a certain security in "if this vendor quits, I can go with another vendor and keep my software and peripherals." There were competitors to Microsoft's MS-DOS even before Linux, you could get disk drives and such from multiple companies...only the CPU was truly proprietary to one company.

So as big businesses and governments started adopting these things and paying BIG bucks investing in computer infrastructure, hardware, software, personnel training etc. a lot of bigwigs started worrying about Intel's future. What if this company goes bust, has a fire at a factory, puts out two whole generations of products that destroy themselves or whatever. Will that pull a rug out from under us? So Intel had to give AMD a license to manufacture x86 chips as a second source.

Add in a mention of Syrix here, a little company that sprung up also manufacturing x86 chips around the Pentium era who didn't have a license from Intel, they reverse engineered and then sold non-infringing compatible CPUs, so there was briefly a third horse in that race.

AMD has served several different roles in the space; they've sold identical copies of Intel chips to the point they had both the Intel and AMD logo on them, they sold low-tier budget options, and on occasion they've actually out-done Intel at their own game.

~~They reverse-engineered them~~

Edit: Huh, apparently I misremembered