I want to ask a technical question about linking to other sites. For example I am linking to 5 sites from my site then I will place link like this right? But if I put a link to my own site's specific URL and through cPanel I redirect that URL to my client's site, like this in cpanel I redirect /outgoinglink.html to www.website.com Is there any benefit in using this "redirect" technique?
I think the use of redirect to outgoing link won't give benefit from your site in terms of PR but can give referring traffic to them.
Correct, there is no advantage other than tracking the traffic which you send to the target page. These types of URL's would typically be used for affiliate systems or URL shortening services.
I think if I don't put direct links to 5 other sites and put a "Redirect" then search engines will see less outbound links because all links are going to my own site's URLs and from there they are redirecting. Or I am wrong here?
They will still be counted as outbound links, just not from the source page but the redirect page. All you would be doing is moving the outbound from one place to another within your web site structure. If you are concerned about the number of outbound links or loosing juice to the outbound links then just add the nofollow tag to them.