Background:
Mapping applications split up data into "tiles" so you can download only the data you are currently looking at. For example, you don't want to download the entire planet, just to look at your own neighborhood.
Historically, these tiles were literally images that the client application (i.e. web map) could "tile" side by side to cover the part of the map you were looking at. Now we refer to those images as "raster" tiles, to differentiate them with "vector" tiles.
Rather than a rendered image, Vector tiles contain the raw data that you could use to render such an image. Vector tiles allow a smaller file size and more flexibility. For example, with vector tiles you can crisply render at partial zoom levels, keeping lines sharp and avoid pixelating text. The client can also have customizable styles - hiding certain layers or accentuating others from the vector tiles.
Vector tiles are not new technology. For example, Google Maps started using them over a decade ago. So why has it taken so long for OpenStreetMap.org? One reason is no doubt a lack of engineering capacity. There were also concerns about older and less powerful clients hardware not being up to the task, but that concern has lessened over time.
OpenStreetMap also has some unique requirements. It is a community edited database, and users want to see their edits soon (immediately really). It's not feasible to dynamically generate every tile request from the latest data, so caching is essential. Still, to minimize the amount of time tiles will be stale, a lot of work went into being able to quickly regenerate stale tiles before the new vector tiles were rolled out.