Broadly speaking, people consume content, and then think about that content. This, combined with experience, results in ideas. Ideas are fleeting, so you attempt to track them somehow (e.g., writing them down). Ideas are related to each other, so you try to capture those relationship as well (e.g., by also writing them down). As you continue this process, your "externalized ideas" grow in volume, and as a result, it's harder to find ideas, and relate ideas to each other. This is a meta-issue resultant from the externalization of your ideas. I.e., you've written a bunch of stuff down and now you need to figure out how to sort it all in a way that is beneficial to you. You need to manage your externalized ideas. A community has built up around this management issue, and they refer to the topic as "knowledge management". (Brief aside: it's a bit bold to claim knowledge, I'd prefer if the space was called "idea management", but so it goes.) To help you address this knowledge management issue, "tools" have been created, whether they be systems of organization, ways of writing notes, or software. These tools are referred to by the community as "tools for thought". In conclusion, tools for thought are tools to aid in knowledge management, whether as a group, or as an individual.
The presumed upsides of good knowledge management is better ideas. One will better suss out relationships between ideas, and perhaps find relationships between their externalized thoughts that didn't originally occur to them internally. That is, your "external brain" helped you find relationships between ideas that your "internal brain" didn't discover on its own.
The software tools in this space that I am familiar with are what I would describe as "personalized Wikipedias". They are software tools of user-generated content and a backlinking system, much like hyperlinks in HTML (aka, the blue links to other pages on the internet). These tools will let you form a "web" of your own thoughts. They will also include an index of some kind, so you can observe textually similar content, even if it didn't occur to you to form a manual (back)link between those two pieces of content (this can be thought of as "discovery" in some sense.)
- Andy Matuschak is a tools for thought personality
- Tiago Forte is a productivity personality
- zettlekasten is a system of knowledge management
- Obsidian is a software tool
- Roam is a software tool