As P.-L. Assoun remarks, Freud's analysis is centered on two phobias, which are considered most significant. They illustrate the general mechanism of phobias, and point to an underlying fear to which all other phobias should be able to be reduced: “agoraphia or 'the fear of spaces' on one hand, and zoophobia or aversion for animals, on the other” (Les phobies, 15).

To discuss zoophobia, Freud relies most often on the case of little Hans, a five year old boy, whose father knew Freud beforehand and who suddenly becomes afraid of horses, to the extent that he refuses to go out in the streets any longer. In fact, the case of little Hans illustrates so well Freud's theory that it could have been solved a priori. Freud knew that a little Hans would come:

I asked Hans jokingly whether his horses wore eyeglasses, to which he replied that they did not. I then asked him whether his father wore eyeglasses, to which, against all the evidence, he once more said no. Finally I asked him whether by “the black around the mouth” he meant a mustache; and I then disclosed to him that he was afraid of his father, precisely because he was so fond of his mother. It must be, I told him, that he thought his father was angry with him on that account; but this was not so, his father was fond of him in spite of it, and he might admit everything to him without any fear. Long before he was in the world, I went on, I had known that a little Hans would come who would be so fond of his mother that he would be bound to feel afraid of his father because of it; and I had told his father this. (Two Cases, "Little Hans", 2031)

After the analysis, the story appears to be simple enough. The phobia was probably triggered by an accident which happened in front of the boy's house, when a horse fell and died. Because of his love for the mother and the oedipal structure, the boy wished his father would fall in the same way but could not formulate it. In fact, he became afraid that the father would punish him for this death wish. Fathers in Freud's world only punish their sons in one way, which is castration. Little Hans came to fear that his father, now identified with a horse, would bite him.

Of course, there were many horses in Vienna at that time, and Hans was genuinely afraid: he could no longer go out. At times, he feared the horse would come into his room. Nevertheless, even in this situation, the phobia has advantages. There is a gain. The boy loved his father, and at the same time feared him, so that there was a conflict which the metaphor of the horse eliminated. It is then easier to avoid horses than one's father. For instance, there should be no horse in the house, whereas the father may be around. That is the point of phobia: an anxiety that could invade one's whole life crystallizes on an object that is not always around:

Now the castration anxiety was directed to a different object and expressed in a distorted form, so that the patient was afraid, not of being castrated by his father, but of being bitten by a horse or eaten by a wolf. This substitutive formation had two obvious advantages. In the first place it avoided a conflict due to ambivalence (for the father had been a loved object, too), and in the second place it enabled the ego to cease producing anxiety. For the anxiety belonging to a phobia is conditional; it only emerges when the object of it is perceived, and rightly so, since it is only then that the danger-situation is present. There is no need to be afraid of being castrated by a father [...] If he is replaced by an animal all one has to do is to avoid the sight of it — that is, its presence — in order to be free from danger and anxiety. (Inhibitions …, 86)

Concerning agoraphobia, or the fear of spaces, there is no such typical example like little Hans. But the story is quite similar. At bottom, there is a repression of a desire (as little Hans first repressed his desire to see his father fall and die like a horse). The subject represses some unbecoming desire. Nonetheless, the "I" is afraid to be punished for its desires. The underlying fear is similar to the little boy's. It is castration by the father, some kind of devouring. Of course, the fear cannot be expressed in this way. It associates with some object in relation to the desire: it could be the place, the open space itself, where the desire could be accomplished:

The agoraphobic patient imposes a restriction upon his ego so as to escape a certain instinctual danger, namely, the danger of giving way to his erotic desires. For if he did so the danger of being castrated, or some similar danger, would once more be conjured up as it was in his childhood. I may cite as an instance the case of a young man who became agoraphobic because he was afraid of yielding to the solicitations of prostitutes and of contracting a syphilitic infection from them as a punishment. (Inhibitions ..., 88)

The same seems to go with the fear of space, of all kinds of space, not just public space, but the space under a window, for instance. In a letter to Fliess, much earlier, Freud mentions another case:

I may not yet have told you about the analysis of several phobias. "Anxiety about throwing oneself out of the window" is a misconstruction by the conscious, or rather the preconscious; it relates to an unconscious content in which "window" appears, and can be dissected as follows:
Anxiety + . . . window . . . and be explained thus:
Unconscious idea: going to the window to beckon a man to come up, as prostitutes do:
Sexual release arising from this idea
Preconsciousness: repudiation; hence, anxiety arising from the sexual release.
Out of this content only window becomes conscious because this element is raised as a compromise formation by virtue of the idea of "falling out of the window," which fits in with the anxiety. So they perceive anxiety about the window and interpret it in the sense of falling out; and even this is not always consciously present. Incidentally, either motive results in the same behavior: they do not go to the window. (To Fliess, December 16th, 1896, 217)

In Vienna, as in Paris (there are numerous instances in Maupassant's short stories) prostitutes called their clients from windows. Freud's patients, who are not prostitutes, might desire to beckon men from the window as a prostitute. The desire is repressed but the image of the window remains associated with an anxiety, the fear of doing something wrong and suffering because of that. The subject tries to find a reason for this anxiety in relation to windows, and ends up thinking of being afraid of falling through the window. But it is not in reality what the fear is about.

At the time of his letter to Fliess, Freud has yet not yet devised his theory of the unconscious, and then he will modify it many times. For instance, he will change his mind on the question of whether the anxiety comes from the “it” or from the “I”. But it does not matter here. Like the agoraphobia of the young man who fears yielding to his desire for prostitutes, the fear of windows comes from a fear of temptation, which crystallizes on some outer object, or space, which is the locus of temptation and becomes the place of punishment.

This displacement also has the advantage that the subjects do not have to wait for the punishment. The very suffering that is produced by the phobia, the fear itself, and the limitations that it brings to their life is a punishment by itself. In fact, another ingredient in the fear of heights could be the desire to destroy oneself, before which the subject then recoils. Windows, or towers, any height, being from the start a locus of temptation, becomes not only the place of punishment but the seat of another desire, that of punishing oneself, through self destruction:

In that case the instinctual demand before whose gratification the ego recoils is a masochistic one; the instinct of destruction directed against the subject himself. Perhaps an addition of this kind explains cases in which reactions of anxiety are exaggerated, ineffectual or paralysing. Phobias of heights (windows, towers, precipices and the like) may have some such origin. Their hidden feminine significance is closely connected with masochism. (Inhibitions …, 164)
 

In any case, the fear at the root of phobias seems to be a fear of being punished by a mythical father: because of one's hostility towards one's father (like little Hans) or because one could yield to sexual desires. It is not expressed in this way in the experience of the I. Precisely, the phobia has the advantage of expressing the fear, and of punishing the subject, through the displacement of the object of the fear. That is why there is such a variety of phobias. The apparent object of the phobia does not matter. It may even come from an erroneous interpretation of the conscious subject, who may be afraid of windows and just imagines that she is afraid of falling through windows whereas she is really afraid of beckoning men from the window. However, underneath this apparent variety, the real fear, the expected punishment, is always the same:

The anxiety felt in animal phobias is the castration anxiety of the ego; while the anxiety felt in agoraphobia (a subject that has been less thoroughly studied) seems to be a fear of incurring sexual temptation — a fear which, after all, must be connected in its origins with the fear of castration. As far as can be seen at present, the majority of phobias go back to an anxiety of this kind felt by the ego in regard to the demands of the libido. (Inhibitions..., 54-55)