you asked 2 different questions :
1) how to upload a PHPBB forum on your site (as mentioned on the topic)
2) how to upload a phpbb forum template on your website (as mentioned on your post)
1) depends on your host, if it allows it or not (bandwidth, space, authorized or not... (many reasons)
Best way as mentioned above is trough FTP.
For phpbb you need to upload ALL the phpbb folder to your host with your FTP (see link to help you connect) than you will need to CHMOD certain folders to allow some files to be read, write or not...
Once chmoded you need to go to install/install.php or install/ it self and run the install ->
That is where you need MySQL, its basically a database where you store certain informations, phpbb needs a mysql database to store infos such as members, profiles, posts, commands... many many things

step by steps you will go through all of it, oh well go to the /doc folder and read the readme / install file

its really simple.
2) assuming you installed phpbb correctly there will be a folder called "style" thats where all your styles are, if my memory is good tonight i think there is prosilver and subsilver already installed inside that folder... what you need to do is find a template (WITH SAME VERSION !! careful for the version, or the template might not work) get this template, unzip the file and put it next to the other templates inside the /style folder than :
go to the acp (admin control panel) assuming you are the admin of the forum and hit on the to STYLE its gonna bring you to a page with a list of the styles, one of them (the one you should uploaded) (or 2 since you might not have installed both of subsilver / prosilver) will have INSTALL written next to it... just hit install and its installed !
now either go to your profile and modify the style or go to your forum options and disallow style changing by users and set the style you want to change

Extra Tip : if you want to modify a style, you need a little of html knowledge and need to modify the overall_header.html and overall_footer.html (located at style/YOURTYLE/template) that is where all the codes of the look of your style is located add, of course also the style.css or stylsheet.css if you want to mod prosilver. .CSS files are located at style/YOURSTYLE/theme.
Hope you got it right ?