I am decidedly of the opinion that in very many instances we can trace such a necessary connexion, especially among birds, and often with more complete success than in the case which I have here attempted to explain.

This quote is suggesting that it is often very easy to understand the connection between different types of bird, and in some cases even figure out why they are linked. The speaker is saying that it is possible to trace the connection between birds, and that it is possible to do this more successfully than in the example that the speaker was explaining.