Apply to become a translator
If you want to provide your own translations, you have to post a request at https://make.wordpress.org/polyglots/ stating which languages you want to translate yourself, so this implies that you actually speak those languages fluently. This is what I have done and what I will describe in this answer.
But if you want to check in translations provided by a third party professional translation service for many languages that you do not necessarily speak and understand yourself, you should instead post a PTE Request to manage a team of translation editors for your plugin.
Be aware of language variations
Often there are several variations of a language. Apart from the regional variations, like de-at for Austria, there might also formal and informal styles, like de/formal using the pronoun “Sie” instead of the informal “Du”. I have been approved to provide translations for both formal and informal German, but not for Austrian and Swiss variations. So make sure to check which versions are available and be precise in your request for approval, to prevent unnecessary further enquiries.
The language variations are optional. If there is no language pack available, then the original string will simply remain unchanged, so usually you will not need to submit translations for British or Canadian English.
Join a WordPress #polyglots channel on Slack
After you have been approved by an authorized member of the WordPress polyglot community, you will the address of an appropriate discussion channel and an email alias at chat.wordpress.org to use there. In my case, this was the German WordPress developer slack team where there is a channel #polyglots for discussion about particular translations. This is how members of the polyglot community can contact you for further enquiries.
Localize your plugin content
Use gettext to translate your plugin’s content like described on developer.wordpress.org .
Localize meta information found in readme.txt
On my plugin page at wordpress.org, when logged in with German language settings, there is a notification stating that my plugin has not yet been translated to German, followed by a link to “help translate it”, which opens a page with translations provided, pending or missing for all languages currently known to the WordPress community.
When clicking on a language you have been approved to translate, you will find a web form where you can enter translated text and localized links to replace your existing content, which is always thought to be in the default language, which is English (US).
You can use the site’s breadcrumb navigation to switch the sub-project. While Development (trunk) (or your respective release branch) enables your to translate the short description etc., switch to Development Readme (Trunk) to translate the rest of your readme.txt.
Await approval
After adding a translation, this translation is still pending, and awaits to be approved by a member of the polyglot community.