Why do we need to amplify DNA sequences?
This is a question I also remember wondering about when I was younger in school. Now as a professional it's way too obvious to even explain. But i think it's an important and common question, which warrants an example or two from common daily lab practice.

Preface

You have to understand that DNA is a molecule. It's really tiny. It's not trivial to work precisely with it, nor detect it or discriminate two different pieces of DNA from each other, let along sequencing them. We make a lot of DNA through amplification so that we can prepare a sufficient amount to make reactions with. 

Visualization

We usually make DNA visible by using chemical dyes that emit light (fluorophores) or incorporated radioactive pieces to make the DNA visible.

Here's a common case in the lab: we need to get enough DNA to use in subsequent reactions. When we work with DNA it is very rare that we obtain exactly what we want 100% of the time, or perfect yield. For example, we want to cut a piece of DNA once. But we have scissors (i.e. restriction enzymes) that will chop up the DNA multiple times, especially when we leave the reaction going too long. What we do is take 100 units of DNA, subject them to the reaction, and then view the results using electrophoresis using an agarose gel, for instance. We can then see 40 units of uncut DNA, 30 units of single cut DNA, and some 30 units of DNA cut multiple times (2+ times, so they are small fragments). We need enough single cut DNA to proceed with the next reaction, with full knowledge that there are no contaminating chunks of DNA. Typically there are many reactions in a row that we do to achieve anything in the lab. Cloning, in vitro amplification, sequencing, gene editing, preparing genes for transgenesis, creating RNAi probes, what have you.

More is better

Another case: if you want to know the sequence of DNA, you must sequence it. However, sequencing is imperfect and errors occur. To be sure, we re-sequence the same DNA many times to get a confident picture of what the sequence is like, beyond reasonable doubt. Sequencing DNA ten times is really more reliable than sequencing it once. This could be because of how the molecules behave in solution, or because of degradation, or tens of other things. You must understand this is also not specific to sequencing; a lot of reactions follow the same logic.

Amplification as a specification tool

By using primers in PCR, we can make sure we are obtaining what we want. In principle, this means that we can make a gajillion copies of something we are interested in (e.g. a gene), even from a single strand of DNA which may contain things we are uninterested in (e.g. the rest of the genome).

Some thoughts

This answer is certainly not extensive, but I hope it's a primer for you to understand that DNA work is just really smart nanotechnology, and we are not yet at the level of performing nanotechnology on single molecules, with full confidence, precision or efficacy. On the daily, we need enough to see it with our eyes so we amplify and react it with intercalating dyes to make them visible to the naked eye!