What's so cool about local compactness?

Every locally compact Hausdorff space is Tychonoff, hence it has "enough" continuous functions. This is used extensively. A prominent example is the proof of Gelfand duality: If we associate to a locally compact Hausdorff space $X$ the $C^*$-algebra $C(X)$ of complex continuous functions, then we obtain an anti-equivalence of categories between locally compact Hausdorff spaces and commutative $C^*$-algebras (with suitably defined morphisms).

Technical but important: If $Y$ is locally compact, then for all spaces $X,Z$ the map $$C(X,C(Y,Z)) \to C(X \times Y,Z)$$ is well-defined and bijective (exponential law). The special case $Y=[0,1]$ shows that a homotopy between continuous maps $X \to Z$ is really just a map from $X$ to the path space $C(I,Z)$.

Quite related to the exponential law: If $Y$ is locally compact, then $Y \times -$ preserves quotient maps (this doesn't hold for arbitrary $Y$, although many topologists use this, perhaps having in mind a convenient category of spaces).