Well, you need a bit more context to really get at this. This comes from The Concept of Anxiety where he wrote:

Anxiety may be compared with dizziness. He whose eye happens to look down into the yawning abyss becomes dizzy. But what is the reason for this? It is just as much in his own eye as in the abyss, for suppose he had not looked down. Hence, anxiety is the dizziness of freedom, which emerges when the spirit wants to posit the synthesis and freedom looks down into its own possibility, laying hold of finiteness to support itself. Freedom succumbs to dizziness. Further than this, psychology cannot and will not go. In that very moment everything is changed, and freedom, when it again rises, sees that it is guilty. Between these two moments lies the leap, which no science has explained and which no science can explain. He who becomes guilty in anxiety becomes as ambiguously guilty as it is possible to become.

So, what he is really saying here is not the pithy statement that anxiety is engendered from freedom. That's just his position leading to the conclusion that the freedom to kill oneself (looking into a yawning abyss insinuates the ultimate freedom) is that basis for all other freedoms, and that it cannot support itself when freedom itself comes to this realization. Our psychology cannot grasp such freedom, so it gets caught in a feedback loop of guilt and anxiety. Because we don't know where the guilt comes from, it is ambiguous and difficult to deal with.

You might extrapolate this to other anxieties and fears, but I think what he is saying here is that anxiety itself is the mechanism that buffers our inability to cope with the understanding that we have an ultimate freedom to destroy ourselves.