By default, the answer is no.
However, you can turn on an option to do this automatically Edit->Catalog Settings->Metadata->Automatically write changes to XMP (or do it manually with Ctrl-S).
Once the keywords (and captions and other metadata is in the files), any photographic application should be able to read them and use them.
Also, if you are going to use some other application in the future, you'd need to Export the photos, and the metadata will be contained in the exported photos as well.
Yes, LR writes the keywords, and indeed all the metadata from the right-hand side (title, caption, capture date, etc.), into industry-standard fields in the image's file. LR is reasonably compliant with industry standards, so you can be confident that your data will be read by programs in the future.
Most people set the option Catalog Settings > Metadata > Automatically Write Changes Into XMP, which will write the metadata back to the file in background after you change it. If you don't set that option, you'll have to manually do Metadata > Save Metadata To File.
Important note for video: LR does not fully support video metadata, and it doesn't write any metadata back to videos. So if you want your metadata to get written back to the file system independent of the LR catalog, you'll need to use either the Video Asset Management plugin or the Any File plugin to manage your video. These third-party plugins create "sidecar" JPEG files into which your metadata changes will be stored.