You can use the archive files to upload to a different server. The base structure should remain and all your content. However, your blog posts will not copy and custom fonts, slideshows and contact forms and the like will probably not work, as they're a part of Weebly's system. Any editing of your site from then on would need to be done traditionally with a website editor. Also, see note, below about the copyright of the weebly theme.
Re-build your site at another template-based website service, your Weebly archive can still be very useful . . ..
Files: This folder contains your site's CSS (Cascading Style Sheet) file(s) and a Theme file folder.
CSS info includes the site dimensions, text and background colors, font preference, sizes and weights, link colors, nav bar setup, etc., which can save you some major rebuild time.
If it's too complicated to figure out, you can take a screenshot* of your site's pages online for reference and use a free color meter** to copy down the colors of your text, background etc..
The Theme folder contains the base images of your previous template, including button files. You won't have the rights to the header photo if you used Weebly's, but if you uploaded your own, it'll be in this folder.
If you designed your own theme images and uploaded them, they're there for you to re-use. If you didn't, and you want to design your own in Photoshop, etc., you can use these images' dimensions to fit perfectly into the layout.
*** Note: If you would like to use the theme for your new site, be sure to check on the rights for the template design, first. (you can see the info by using your browser to "View Source." ***
Uploads: click into the nested folders until you reach your files. Everything that you uploaded to your site will be in here (images, pdfs etc.). They may not have the same name as when you put them up, but they can be used again for your new site.
Html Files: While some of the code won't be useful, double-click on the html file while you have your browser open and you'll see the page as it was at the time of archiving, which is a great help in rebuilding. With this, you will be reminded of the layout. You can also highlight the text and copy it for your new site.
If this doesn't work for you and you're really in love with the look and organization of your present site, be proactive and take screenshots* of your site's pages online for future reference.
IMPORTANT, Blog Posts: If you have a blog page and want to transfer its content, copy and paste the the posts from your site online into a word processing document and save them as a text file to re-post later. Otherwise, they'll be lost.
*On a Mac, press Command/Shift/4 to select a portion of the screen. (Press the Esc button if you change your mind and don't want to take a screenshot, or if you got your crops wrong and want a re-do.)
**DigitalColor Meter: Macs come with a free app called DigitalColor Meter, which can be found in your Applications or Utilities folder. Select the "RGB as Hex Value, 8-bit" setting and use its eyedropper tool to sample colors you might like to use. You can sample from any image on your desktop, including images you have on your site and anything you can view on a browser.
The hex code will appear in DigitalColor Meter's three windows. Copy down the number pairs in order (R first, G next, etc.) and add a hashtag (#) at the beginning when you enter the code into the white window at the bottom of the swatch palette. For example, the coral subheads on this page are R=C4, G=72, and B=69, hence the hex code is #C47269.
Eyedropper: I've heard there's an app which works on a PC, called Eyedropper, but have not used it.
***Some themes have been donated to Weebly by designers and will have their contact info in the source code. To find out if your template has this info, go to your published site online, right-click on the page and select VIEW SOURCE. For my fellow Mac users, control-click on the page and select VIEW PAGE SOURCE. If you are using Dreamweaver, check out the html code.