Definition of connectedness? Connectedness is defined as: "A metric space $E$ is connected if the only subsets of $E$ which are both open and closed are $E$ and $\varnothing$. A subset $S$ of a metric space is a connected subset if the subspace $S$ is connected."
Can someone provide me with a more trivial/simple definition of connectedness?

The definition is what it is. There is no "more" or "less" trivial definition, since they are all equivalent.

That said, you may want a more intuitive explanation of connectedness.
Your definition is for example equivalent to saying that $E$ is connected if there do not exist two open sets $U,V$ such that $U\cap V=\emptyset$ and $U\cup V=E$.
The explanation is that intuitively,

A metric space is connected if it cannot be split into two pieces without "tearing" it somewhere.

