There are multiple symbols commonly used to represent "such that"

1. \backepsilon or \ni
2. \mid
3. :

For "therefore", I usually see \therefore

As stated in the comments, you get the symbols in mathmode simply by writing them down. Packages like amsmath and amssymb support you.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\newtheorem{theorem}{THEOREM}
\newtheorem{proof}{PROOF}
\begin{document}
\begin{theorem}
If an operator has both Left Identity and
Right Identity then it is \emph{UNIQUE}.
\end{theorem}
\begin{proof}
Let $e_{l}$ is left identity 
$\therefore e_{l} * e_{r} \implies e_{r}$
\end{proof}
\end{document}

"It seems like $\therefore$ is more frequently used when reaching the conclusion of an argument, while $\implies$ (alternatively $\rightarrow$) is for intermediate claims that imply each other."

Your supposition is largely correct; my only concern is your description of $\implies$ being used to denote intermediate claims (in a proof or an argument, for example) that imply each other. The $\implies$ denotation, as in $p \implies q$, merely conveys that the preceding claim ($p$, if true) implies the subsequent claim $q$; i.e., it does not denote a bi-direction implication $\iff$ which reads "if and only if".

'$\implies$' or '$\rightarrow$' is often used in a "modus ponens" style (short in scope) argument: If $p\implies q$, and if it's the case that $p$, then it follows that $q$.

Typically, as you note, $\therefore$ helps to signify the conclusion of an argument: given what we know (or are assuming as given) to be true and given the intermediate implications which follow, we conclude that...

So, put briefly, $\implies$ ("which implies that") is typically shorter in scope, usually intended to link, by implication, the preceding statement and what follows from it, whereas '$\therefore$' has typically, though not always, greater scope, so to speak, linking the initial assumptions/givens, the intermediate implications, with "that which was to be shown" in, say, a proof or argument.

Added:

I found the following Wikipedia entry on the meaning/use of the symbol'$\therefore$', from which I'll quote:

To denote logical implication or entailment, various signs are used in mathematical logic: $\rightarrow, \;\implies, \;\supset$ and $\vdash$, $\models$. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign $[\;\therefore\;]$ is traditionally used as a punctuation mark, and does not form part of a formula.

It also refers to the "complementary" of the "therefore" symbol$\;\therefore\;$, namely the symbol $\;\because\;$, which denotes "because".

Example:

$\because$ All men are mortal.
$\because$ Socrates is a man.
$\therefore$ Socrates is mortal.

If you add \usepackage{amssymb} to your preamble (or any package which loads this), you can just write \therefore in maths mode. As Willie Wong says, if you have even a minimally functional installation of LaTeX, you already have this package.

If you are using non-default text fonts, you might want to use a different set of fonts for maths, too, but \therefore will be available in any reasonably comprehensive configuration.