An e-mail client is essentially a program that you install locally on your computer. It can either synchronise with an online e-mail storage or download e-mails for local storage. In your case I would install something like Thunderbird (freely available:
https://www.thunderbird.net/en-GB/) and setup your existing account as a POP3 connection so that all the e-mails from your 'old' provider will be downloaded to your computer and will be stored locally.
That's not quite what you want, as you will not be able to access them online from then on (or you will only see the new ones there), but I think that's what your provider is asking you to do, as the online storage used will be a lot less from then on. I'm sure someone else will be along shortly with other suggestions. Migrating them all to a new online storage is not something I can help you with, but perhaps there are solutions that will do that.