Why do we need deep sequencing?
Short Answer

In a nutshell, DNA sequencing technology has a limit to how long a stretch of DNA it can read in one go.

Long Answer

So what most commonly occurs is the length of DNA you wish to sequence needs to be (almost randomly) chopped up into given lengths (depending on the technology) and each length or read is sequenced in parallel. But now you don't know which lengths to stick together! i.e. the ordering of these chopped up lengths. So you sequence it many times over (deeper) and this way you can align contiguous ends of these lengths together like a jigsaw puzzle. The more times you sequence the more accurately the 'contigs' will be aligned.

DNA Assembly Diagram



Comment if you want more details, but the technical term you want to google is "sequencing assembly". 

Here's the wiki 

:)