How to edit when "Animation Explosion" effect takes place?

I assume that you are talking about the Explosion animation preset that is found in the Presets>Miscellaneous folder.

When you apply that preset a starting and ending keyframe for range selectors in 3 Text animators are applied to the text layer. To see everything that has been done to the layer, select it, press the 'u' key twice to reveal all modified properties and start moving keyframes and properties around to make changes to the look and the timing of the preset. This should be standard procedure when you are having trouble animating any layer. Press 'uu' to reveal everything you have changed and make adjustments. If necessary, solo the layer and turn on and off effects.

By lyric video I'm assuming that you are animating text to music or dialogue. There is a tendency for rookies to try and do an entire song in a single comp. The right way to work is to break the audio track into sections that are no longer than a sentence or phrase and treat each as a separate composition. This keeps the comps manageable, makes fine-tuning each phrase or sentence much easier, and even though it sounds like a lot more work, it is actually a lot faster way to do that. Depending on the project you can then either render each comp to a suitable production format (not an mp4) and edit the final in Premiere Pro so you can make any needed adjustments, or you can just sequence all of your nested comps in the main comp and render the whole project in After Effects. That will help your preview problem a great deal. 

There are other preview workflows that you should follow. I use the "pencil test, ink and paint" workflow cell animators have been using for more than 100 years. The "pencil test" is a low-resolution preview with no effects or motion blur to check the timing and action in the scene. When the "pencil test" is ok and things flow as they should I'll turn on motion blur and add or turn on effects, set the comp panel resolution to Full and look at a few critical frames to check them for color and accuracy. When the critical "hero" frames are good I send the comp to render and move on to another part of the project. I only run a full resolution full effect preview on more than about 10 or 15 frames on about 10% of my comps. If a comp is longer than about 7 seconds I almost never run a full resolution full-scale preview because it is a waste of time and I already know what the critical parts are going to look like from the "ink and paint:" tests.  The more experience you get the more you will know what things are going to look like without waiting for full resolution previews of the whole comp.

I hope this helps. If you are new, please spend a fair amount of time looking at the tutorials in the User Guide. They will teach you how the UI works and explain some of the tens of thousands of possible workflows you can use in After Effects.