From chapters 2-3, we gather that at least some of the churches to whom Revelation was written were facing a good deal of persecution. In addressing their situation, one of the main themes of the book is that perseverance in trial, even to death, is the manner in which the churches will "defeat" their enemies (along with the word of their testimony).
We see this, for example, in the sections addressed to the individual churches. To the church in Smyrna, he writes (v 2:10), "Do not be afraid of what you are about to suffer. I tell you, the devil will put some of you in prison to test you, and you will suffer persecution for ten days. Be faithful, even to the point of death, and I will give you life as your victor's crown." We see here the paradox: being faithful to death results in the victor's crown that is life. The language is put in antithesis in order to provoke thoughtful response.
Or again we find the theme in the song of chapter twelve. In 12:11, it is said, "And they have conquered him [the accuser] by the blood of the Lamb and by the word of their testimony, for they loved not their lives even unto death." Victory is won over dragon who is cast down from heaven because of the faithfulness of the saints in their witness - faithfulness even to death.
Similarly here with the Lion/Lamb imagery, this theme is reinforced by the exemplar par excellence. The Lion of Judah conquered by becoming the Lamb who was slain. John ties the two images together in order to draw out this theme. The Lamb stands as the chief witness to the pattern being put forward: that one conquers as a saint not by taking a life, but by laying one's own life down.