The augmented-sixth chord does not resolve correctly, instead shifting to a chord progression that fits best in the key of A minor: iiø43 - viio42 - V7. By half-steps the dominant chord gets transformed, leading us back to the key of E minor. A minor is hinted at several times, and the final cadence of each phrase (there are only two phrases in the 25-measure prelude) includes an oscillation between the dominant B7 chord and the A minor triad.
The second phrase (measure 13) repeats the first phrase for the first 1.5 measures. At that point a subtle shift in the underlying chords, merely by moving the bass note one beat earlier, creates new tensions in the simple melody. These tensions lead to a passionate cry as the melody tries to escape the tyranny of the half-step. It leaps up to a higher register and throws in large intervals of 6ths and 7ths. But the plodding chords rein in the melody, pulling it back to a simple two-note pattern and soft dynamics.
This prelude is all about the tensions between the melody and the harmony, with the harmony clearly winning. But what is so striking is that the exotic harmonies are created by simple means, small little movements of the left hand, and this slow harmonic rhythm creates such emotional intensity. John Sloboda and Andreas Lehmann picked this piece for a study on the perceived intensity of emotion, because it is short yet full of possibilities for interpretation. James Huneker puts it well in the introduction to the Schirmer edition:
The whole is like some canvas of Rembrandt -- Rembrandt who first dramatized the shadow in which a single motive is powerfully handled; some sombre effect of echoing in the profound of a Dutch interior, all gold and gloom. For background Chopin has substituted his soul; no one in art but Bach or Rembrandt could paint as Chopin did in this composition.
For a real brain twister, try to analyze the second prelude!