Mathematically, the only difference between convolution and correlation is that one of the inputs is reversed for the convolution. This reversal is necessary to perform the "filtering" operation that MBaz was referring to. In performing a correlation, no time reversal is needed, as you are just trying to perform a pattern match between two inputs.
To intuitively understand what the correlation is doing, consider the following example. Your friend took a picture while on vacation in some foreign country and sent it to you, but failed to mention where the picture was taken. You decide that you want to identify the location that the picture was taken by traveling to that country yourself. Wherever you travel, you hold that picture up to the scene that you are viewing and are looking for similar features between the picture and the scene. The more similar the features of the picture and the scene, the more confident you are that the picture was taken where you stand. The correlation performs this comparison by sliding one input (i.e. the picture) over the other input (i.e. the scene) which is similar to you traveling to different places, and for each position the picture is in, performs a multiplication of all pixels between the translated picture and the scene before taking the sum those products. The result is a measure of confidence of the picture being taken in that position.
Can it only detect the presence of original signal or can it also
reconstruct and recover the original signal?
The correlation makes no assumptions about what the inputs are, nor whether one input was corrupted in any way, so asking about signal reconstruction without any further context does not make much sense. You are simply comparing two inputs over some time/position offset.
"If the correlation is large for a given time delay τ, then we may be
confident in saying that the answer is yes"
MBaz simply states that the larger the output of the correlation for a given time delay/position, the more similar the picture and the scene were to each other. It is up to the user to interpret what the exact magnitude of the output of the correlation means, as a high output could mathematically originate from two situations - your inputs are very similar to each other (a good match between picture and scene), or the scene has a localized high amount of energy relative the rest of the scene that causes the multiplication-sum operation to be larger than the other positions/time delays. Also remember that just because the picture and scene may be similar does not mean that they are the same, which could be a potential false positive detection in the pattern matching sense.
