I have set up the nodes shown in the image. I wanted to transform the big sphere when the small sphere comes near it. I can visualize that the attribute is being transferred. But it does not actually transform the big sphere.

1.) Does the attribute transfer not work with the transform node?
2.) Have I connected this wrong way?
3.) what does the attribute transfer return?

It can only transfer where the vertices meet other vertices. What you can try is to drop down an IsoOffset right after your smaller sphere. Follow that with a Scatter node and increase the count so the entire volume of the small sphere is filled with points. Having more points helps attributeTransfer detect when a nearby point is available to supply new attributes.