AMD being competitive means faster and / or cheaper CPUs for everyone, short term from AMD and medium term from both AMD and Intel. People have been frustrated with slow CPU performance progress resulting from diminishing returns of process improvement and lack of competition for Intel ("four cores should be enough for everyone and also expensive").
For most developers, it doesn't. They're both making x86_64 chips with coherent caches of similar size, similar core and thread counts, and a lot of overlap in instruction set extensions. For people doing very high performance work (heavy data processing, simulations, etc.) or performance critical work (think hand coding optimized crypto library routines in assembly), the subtle differences in those categories might affect how they lay out data and access it, how many threads they use and how they use them, and preferences for the availability of instruction set extensions that are specific to the kind of workloads they have (vector processing extensions, native crypto operations, unusual bit twiddling patterns).
>For most developers, it doesn't. They're both making x86_64 chips
This is true for the vast majority, but there are niche cases were there are differences even though both are both x86_64(like Intel's FlexMigration and AMD-V Extended Migration).
Competition benefits the space. It lowers price of compute and advances technology. One party prevailing over the other for a significant length of time leads to monopoly economics and stagnation of the industry.
The context of recent news is that Intel has been dominant for many years, prices of processors have been high and growing, and performance improvements minimal. AMD is now offering cheaper and faster processors, which characterises a resurgence of competition in the desktop and server compute market. Ideally Intel will improve their offering in response, otherwise facing loss of the market to AMD, and further continue the competition. No impartial consumer should want either company to prevail, but AMD's present lead signals an end to Intel's dominant position.
Developers benefit directly from cheaper and more plentiful computing, but also indirectly as more applications and approaches become practical for their target platform. As devices become cheaper, the potential user base for an application becomes larger.
Developers will be more indirect beneficiaries since the cost of compute may go down since they will have leverage against Intel and even AMD. Other benefits may include faster refresh cycle and/or hardware bugs effecting the dominant chip maker does not have to drag down the entire ecosystem etc.
Ignoring who is "winning" - it's always good when new viable competition enters a previously dominated market. In this case both companies will be providing to increase performance and lower cost, which AMD has done a great job at - this forces intel to lower prices or improve features to stay competitive.
Win-Win for consumers either way. And for the last what almost 10 years Intel has been ahead in many use cases - and their new offerings are rather stale IMO.
It depends where on the stack you are doing development. For compiler developers, kernel developers and driver developers it could have a reasonable affect. For user-space application developers and web developers it should have very little affect.