As user mtcs posted below, this one-liner will grab the necessary packages without grabbing non-Fedora/RedHat packages (which my previous instructions did, causing needless headaches):
> yum install ftp://linuxsoft.cern.ch/cern/updates/slc6X/${ARCH}/RPMS/php-pecl-uploadprogress-1.0.1-1.slc6.${ARCH}.rpm
Next, restart Apache / httpd to have PHP load the UploadProgress extension:
> service httpd restart
Finally, test by going to /admin/reports/status on your site to see if UploadProgress is reported to be working.
OpenVZ VPS / PHP 5.4.11
if you indeed use the latest PHP version from remi repository and install uploadprogress rpm you may end up with incompatible PHP/uploadprogress API versions. If you run a CentOS 6 openVZ VPS you should install first the relevant kernel headers from openvz.org, then install php-devel 5.4.11 and php-pear from remi repository, and finally build/install uploadprogress via pecl:
> uname -rm (e.g.: 2.6.32-042stab065.3 x86_64 )
> wget http://download.openvz.org/kernel/branches/rhel6-2.6.32/042stab065.3/vzkernel-headers-2.6.32-042stab065.3.x86_64.rpm
> yum --nogpgcheck localinstall vzkernel-headers-2.6.32-042stab065.3.x86_64.rpm
> yum install gcc-c++ php-devel php-pear
> pecl install uploadprogress
> echo "extension=uploadprogress.so" >> /etc/php.ini