This statement occurs in Chapter 13, when Holmes is convinced that he has the villain Stapleton.
“Exactly. This chance of the picture has supplied us with one of our most obvious missing links. We have him, Watson, we have him, and I dare swear that before to-morrow night he will be fluttering in our net as helpless as one of his own butterflies. A pin, a cork, and a card, and we add him to the Baker Street collection!” He burst into one of his rare fits of laughter as he turned away from the picture. I have not heard him laugh often, and it has always boded ill to somebody. (Chapter 13)
What Holmes is laughing at here is the portrait of Hugo Baskerville, the seventeenth-century ancestor of Sir Henry, during whose time the legend of the hound had originated. Holmes discerns in the portrait of Hugo a strong resemblance to Mr. Stapleton. This gives him a clue to why this man is so bent on destroying Sir Henry and the Baskerville line.
"Ha, you see it now. My eyes have been trained to examine faces and not their trimmings. It is the first quality of a criminal investigator that he should see through a disguise.”
“But this is marvellous. It might be his portrait.”
“Yes, it is an interesting instance of a throwback, which appears to be both physical and spiritual. A study of family portraits is enough to convert a man to the doctrine of reincarnation. The fellow is a Baskerville—that is evident.”
“With designs upon the succession.”
Stapleton, Holmes is saying, not only looks very much like his nefarious ancestor, but is as evil as Hugo was. This missing link provides the key to why Stapleton is perpetrating the fraud of the spectral hound, and with this knowledge Holmes will be able to catch Stapleton in the act of trying to kill Sir Henry. It is the final clue to the mystery.
The next evening Holmes lures the hound out to attack Sir Henry, which has been trained, with Sir Henry's stolen boot, to follow his trail, and when the beast attacks Holmes and Watson shoot it dead. This done, Stapleton flees, leaving behind his wife tied up in his house. Mrs. Stapleton, however, knows where her husband may have fled, and with her help Holmes and Watson discover that he must have drowned in Grimpen Mire.