To export the Twinkle gadget, click on "Download" button, save the downloaded file, go to Special:Import on destination wiki and upload it. Then add the following to MediaWiki:Gadgets-definition page:

* Twinkle[ResourceLoader|dependencies=mediawiki.user,mediawiki.util,jquery.ui.dialog,jquery.tipsy|rights=autoconfirmed]|moment.js|morebits.js|morebits.css|Twinkle.js|twinkleprod.js|twinkleimage.js|twinklebatchundelete.js|twinklewarn.js|twinklespeedy.js|friendlyshared.js|twinklediff.js|twinkleunlink.js|twinkledelimages.js|friendlytag.js|twinkledeprod.js|friendlywelcome.js|twinklexfd.js|twinklebatchdelete.js|twinklebatchprotect.js|twinkleconfig.js|twinklefluff.js|twinkleprotect.js|twinklearv.js|friendlytalkback.js

You must have appropriate permissions on destination wiki (including the right to edit system messages) and import from file uploads must be enabled.