Unlimited bandwidth means that your site can be visited an unlimited number of times by visitors without your site going down/becoming unavailable.
I've used 000webhost in the past with no issues, although they had a data breach a while back. Must've upped their security since then. But my site has never really gone down while using it (which is an issue you can have with free hosts). Then you just use something like Filezilla to upload it.
I don't know if Weebly has the capabilities to upload a site though, unfortunately. At least not in the free version, I don't think.