A good rundown can be found here. Some headlines are as follows:
Though never having studied in traditional yeshivot—being primarily an autodidact he developed a unique Talmudic methodology and was one of the pioneers in what has become known as the Analytical School of Talmudic scholarship...
...Engel was the un-bifurcated consummate gaon who weaved the various streams of Judaic thought into a cohesive picture. He would use agada to explain halakha, halakha to explain agada and bring Talmudic proofs to Kabbalistic concepts1. For him Torah was a singular unite and the divides between sections was artificial. While at first glance the relationship between sections may not be apparent a deeper look—according to Engel—reveals that inherent association...
...While many have the ability to locate the central theme of a Talmudic section, Engel had the ability to see how the seemingly trivial aspects of a Talmudic section were central as well. This was due to his study not only analyzing the section under-discussion but locate the conceptual paradigms that were conveyed in it as they pertain to a general Talmudic clause. Engel did not only know Talmud, he was a master of Talmud and employed the keen ability to manipulate the data and trick out new modes of thought...
One of the central characteristics of Rabbi Yosef Engel’s Halachic methodology is his persistent search for Shoresh HaDin—the philosophical conceptual roots underlying Halachic ideas. In Engel’s Lekach Tov and Atvan D’Oraytha he analyzed many of the same questions tackled in the Lithuanian analytical academies and he discovered and categorized Halachic axioms and principles that had never previously been identified....
There is a lot more that one can glean from the above-cited website which gives a really good insight into his approach.