Why do we study Polish spaces?
You ask a very general question which is hard to answer completely... Here's a loose and unfinished collection of thoughts that might be used for fleshing out a more polished answer.
I suppose one main reason for the interest in Polish spaces is a combination of the simplicity of the concept and the fact that one can achieve reasonable generality and usefulness outside of the realm of descriptive set theory itself.
The definition itself is easy to understand: one barely needs more than a first course in real analysis to have the necessary background to be able to grasp the definition.
The setting is convenient: topological and measure-theoretic pathologies are kept to a minimum. Many objects of interest are covered.
Polish spaces allow for powerful results, quite sufficient for 'everyday mathematics'.
One reason certainly is that Polish spaces have reasonable closure properties, for example, they are closed under taking countable products and passing to open, closed or, more generally, $G_{\delta}$-subspaces. It was also mentioned in another answer that in many fields one never really cares about objects that aren't Polish spaces, so results about Polish spaces are usually 'good enough'.
A further reason is that Polish spaces are "well-behaved": they cannot be 'too large' or 'too intangible', so various 'annoying' pathologies are excluded. The facts that they admit an open surjection from the Baire space and that they can be embeded into the Hilbert cube can be used to reduce questions about general Polish spaces to spaces that one understands relatively well.
Especially the 'universality' of the Baire space allows for powerful, rather explicit, coding techniques, like Lusin and Suslin schemes. These codings can be exploited to give explicit constructions (thus guaranteeing measurability) of sections for surjections, or selections of points in families of sets, which is very useful in various applications where the functions obtained directly from the Axiom of Choice aren't good enough due to their potential lack of measurability.
Surely, the Borel sets of Polish spaces are as important in applications as Polish spaces themselves. In probability theory there is often an explicit or implicit assumption that the underlying measure spaces are standard Borel spaces.
Since you ask about applications, let me mention a few of those.
Outside of descriptive set theory, Polish spaces are ubiquitous: the best-understood objects are Polish: most spaces arising in geometric contexts and well-behaved things like locally compact second countable spaces and groups, manifolds, separable Banach spaces and algebras, and so on. On the other hand, they are an almost indispensable tool for handling 'tame' measurable spaces (standard Borel spaces) in probability theory and ergodic theory.
Probability theory certainly is one major field of application, Polish or standard Borel spaces allow the formulation of Kolmogorov's consistency theorem, the construction of conditional expectations and Prokhorov's theorem in useful generality. Moreover, measures on Polish spaces have decent regularity properties and one has a painless formulation if the Riesz representation theorem. Applications include the construction of sequences of random variables with prescribed distributions and the treatment of discrete Markov processes. See for example Parthasarathy's book on Probability measures on metric spaces, for an exposition of these results that doesn't require much background in probability.
People interested in probability theory, ergodic theory or functional analysis (operator theory and representation theory) probably aren't too interested in Polish spaces as such, but it crystallized over time that many of the deeper results in their domain are quite satisfactorily covered by general theorems on Polish spaces (or their underlying Borel spaces).
A number of famous mathematicians used Polish spaces in part of their work and contributed to the recognition of the concept as central to a non-negligible part of mathematics:
Blackwell argued successfully that many of the pathologies inherent in Kolmogorov's approach to probability theory can be avoided by restricting attention to standard Borel spaces.
Rokhlin accomplished similar things in his thesis on ergodic theory.
Choquet in his work on potential theory (as exposed in the standard treatises by Dellacherie-Meyer or Doob).
Mackey (and later Effros) in representation theory of locally compact groups and Banach algebras and their applications to quantum mechanics.
Zimmer used Polish spaces crucially in his approach to Margulis's theory of lattices in semisimple Lie groups ("super-rigidity").
All these mathematicians not only applied existing results from descriptive set theory, but they also contributed back to the general theory with their own insights (partition theorems, the theory of capacities, selection theorems, etc.). This brief list already shows that Polish spaces are of interest to mathematicians with otherwise rather disjoint focus.
One prominent central result is the theorem on disintegration of measures. Apart from the probabilistic applications linked on the Wikipedia page (e.g. conditional probabilities), it is also a central part of the 'Mackey machine' in representation theory (loosely speaking, the decomposition of representations into a direct integral of irreducible representations) and is used throughout ergodic theory in guise of 'ergodic decomposition'.
To sum this attempt at an answer up: there is a lot of interest from outside descriptive set theory which motivates further study.
The books by Kechris and Srivastava both give a good overview of numerous basic applications to other branches of mathematics while they develop the fundamental machinery.
In more recent days there has been considerable interest in Borel equivalence relations, see e.g. the work of Hjorth and various surveys by Kechris.