What is the advantage of indirect ELISA over direct one?
You are correct, the selectivity advantage of an indirect or sandwich ELISA comes from the fact that two antibodies are employed - one to capture the analyte, the other to detect it.

Here is the classic illustration of how this type of ELISA works. First ([1]), the capture antibody is coated onto the plate and bound via one of a number of different types of chemistry. The plate is then washed (this occurs between all steps). [2], the analyte is added, sometimes in a homogenous solution (i.e., pure recombinant protein), other times in a matrix like serum, cell lysate, etc. The plate-bound antibody has a certain specificity for the analyte &mdash; let's say it binds the wrong epitope one time in 1000. [3] The detection antibody is added, and let's say it has the same specificity of 1/1000. Finally, [4] the secondary antibody (with the enzyme linked to it) is added, and [5] the enzyme's substrate is added, producing the color or light output.



From Wikimedia

If we were to use a direct ELISA, the error rate would be 1/1000. However, by combining two antibodies, our error rate is now 1000 times lower &mdash; 1 in 1 million. Since both antibodies are required to bind properly in order to get a signal, this makes the results much more reliable.