I have came across the following sentence in an instruction manual and I'm curious why they did not use them in the te-form.
この取{と}り扱{あつかい}説{せつ}明{めい}書{しょ}をよくお読{よ}みいただき, 正{ただ}しくお使{つか}い下{くだ}さい。
In formal Japanese, this is the standard 連用形. It's used in place of て form of verbs when linking. It's usage in 敬語{けいご} actually goes beyond this, with these suffixes also helping form verbs. In the context of your quote, both are replaceable with て form in everyday speech.
The general rule being to take the ます stem, remove this, and replace as necessary
座ります -> お座りください Please sit
伝えます -> お伝えください Please tell them
連絡します -> ご連絡ください -> Please contact / get in touch
Note that you remove する if the verb takes this. Other examples for 〜頂く：
お読みいただき
お電話いただき
Not all verbs conjugate like this, and some take ご rather than お. The instruction manual uses them to keep the writing polite and formal to the buyer. As for when to use it yourself, the finer rules are all part of 敬語{けいご}, which is itself more complex than can be summarized in one answer.
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7I'm curious as to why this answer was downvoted. If it's wrong, incorrect somewhere, or poorly written, then please explain why. In the meanwhile, I'll upvote. – dotnetN00b May 22 '15 at 14:09
