It is steep, but at the same time, representation theory is so
damn beautiful; for anyone with the time and interest to pursue a
pure mathematical topic as a hobby, I highly recommend
combinatorial representation theory. Besides, representation
theory seems to be of great importance to computer science. The
first explicit examples of expander graph families, for instance,
were given by Grigori Margulis using deep theorems from unitary
representation theory (see A. Lubotzky, Discrete Groups, Expanding
Graphs and Invariant Measures for instance).

Regarding the sentence you posted, it is a bit less scary than
it sounds. Let's break down the concepts.

Symmetric group of order N: it is the collection of all
rearrangements of N distinct symbols on a line. This collection
satisfies the axioms of a permutation group, which very roughly
means a self-contained collection of permutations.

Representations of... (a group): Abstract groups are opaque objects
in general, but some particular groups tend to express information
vividly in their elements and structure. The most computationally
rich example is groups of linear transformations on a vector space:
rotations, reflections, shears etc. A linear representation of an
abstract group is a realization of the group as a group of linear
transformations: you pick a vector space, and let each group
element correspond to some linear transformation on the vector
space, so that the product of group elements corresponds to the
composition of linear transformations. Now you can use the power
of linear algebra to study your abstract group.

Irreducible representations of... (a group): Now that you have
realized your group as a group of transformations on a vector
space, maybe you get the following behavior: the group
takes a subspace of that vector space and transforms it
into itself. And takes the complementary part of that vector
space, and transforms that into itself. Well, then you can study
the two subspaces separately because they are never "mixed" by the group.
That is called a reduction, and such a representation is called
reducible. If you cannot do that, you have an irreducible
representation. These act like the prime numbers of representation
theory, at least where finite groups are concerned: every
representation splits as a sum of irreducibles, and the deepest
linear information about the group is carried in them.

Branching properties: so now you have an irreducible representation
of a symmetric group. Every group of permutations of N elements is
a subgroup of that group, so what happens if you focus your
attention to the subgroup, and forget about all the transformations
that are not in the subgroup? The resulting representation may not
be irreducible anymore: some transformations that mixed vector
spaces may not be in the smaller group. If it is not irreducible,
how does it split into irreducible representations? In
particular, how many irreducible components are there and how many
times does each component appears? The answer to those questions
are the branching rules for the restricted representation to the
subgroup. In general, it is an extremely difficult problem, but it
has been resolved for some families of groups, the symmetric groups
being a primary example (where the rules are given in terms of,
well, Young tableaux and the linked Young lattice!).