The fundamental idea of a coset is that of taking a subgroup and "translating" it to fill up $G$. A coset is just one such "translate." No coset but $H$ itself is a subgroup of $G$, since they don't contain $1$, but there is a lot of ground that can be covered by treating them like elements of a group or set instead. Since the cardinality of the cosets is constant, they represent ways of dividing up $G$, and you get nice formulas like $|G|=|H|[H:G]$, where $[H:G]$ is called the index of $H$ in $G$ and is equal to the number of cosets.
In my opinion, the biggest use of cosets is to set up the idea of a quotient group. You may have seen the general idea -- formally dividing one mathematical structure by a substructure -- in other fields, like topology. What you do in almost every case is add, to the laws defining your mathematical structure, another law that sets everything in the substructure equal to a point, or the identity, or zero. Then you try to make everything else work.
In group theory, in particular, the set $G/H$ (or $H\backslash G$) always has an induced group structure when $H$ is a normal subgroup. And in fact, it's easy most of the time to just think of its elements as abstract and indivisible group elements. But sometimes, especially when you're proving that this works, it's useful to think of them as equivalence classes instead, where the equivalence relation is of the form $a\sim b$ when $a=bh$ for some $h\in H$. Of course, such an equivalence class is just the coset $bH$.
I'll give one nice example with abelian groups before moving on.
The quotient group $\mathbb{R}/\mathbb{Z}$ is the set of cosets of the form $x+\mathbb{Z}$, with $(x+\mathbb{Z})+(y+\mathbb{Z})=(x+y)+\mathbb{Z}$, which is just the induced addition. But you can also think of the elements of the group as numbers of the form $0\le x<1$, with addition being done "modulo $\mathbb{Z}$": that is, $x+y$ in $\mathbb{R}/\mathbb{Z}$ is equal to the fractional part of $x+y$ in $\mathbb{R}$. So all we really have is a circle with an addition operation. One place where this pops up is the group formed by numbers $e^{i\theta}$, the unit circle in $\mathbb{C}$, under multiplication.
(Another, weirder example is $\mathbb{R}/\mathbb{Q}$. I don't know much about it in terms of group theory, but it gives rise to a standard example of a non-measurable set in analysis.)
Another use of cosets will pop up when you study group actions. I think this is just called the coset action. Basically, given a subgroup $H$, $G$ acts on $G/H$ by multiplication: $(g,kH)\mapsto gkH$. Surprisingly, this action works differently than $G$'s normal multiplication on itself. I don't know any good examples of this.
Now let's look at the coset questions you mentioned.
First, the line $\lbrace 1,a,b,c,d,e\rbrace=\lbrace\lbrace 1,a,b\rbrace,\lbrace c,d,e\rbrace\rbrace$ looks like a typo or something.
Now let's look at the right cosets of $\lbrace 1,a,b\rbrace$ in $G$. If you look at the top left corner of the multiplication table, you see that right multiplication by $a$ or $b$ just gives you the same set $\lbrace 1,a,b\rbrace$, though its elements are permuted. Multiplying by $c,d,$ or $e$ gives something different. We don't have to check that these all give the same coset, since the sizes of the cosets are identical.
In the second example, you can look at the two columns labeled $1$ and $c$. Then just scan down the rows: $a\mapsto\lbrace a,d\rbrace,b\mapsto\lbrace b,e\rbrace,c\mapsto\lbrace c,1\rbrace$, and so on. These are all cosets, and identifying three distinct ones is easy. They also show you something important: if, for example, $aH=\lbrace a,d\rbrace$, you know for a fact that $dH$ is the same set, simply because $H$ contains $1$. Thus, the elements of a coset always give the same coset when multiplied by the subgroup.
Now let's do the same thing for the left cosets of $\lbrace 1,e\rbrace$. Again, ignore everything but the $1$ and $e$ columns, and scan down. We get $aH=\lbrace a,c\rbrace =cH,bH=\lbrace b,d\rbrace =dH$. And, of course, $eH=H$ since $e\in H$.
The right cosets of $\lbrace 1,d\rbrace$ will be similar, but you'll be using the rows $1$ and $d$ instead of columns.
Good luck, and I hope this was at least somewhat helpful!
Also, what book are you using? I learned all this from Artin, but it's possible that you might be getting a different perspective if you have a different book.