Zoning out is common. My tricks: I chant the LOTH in Latin, then read through the psalm silently in French. I also read the psalm title, and the little quote under it from the Bible or a Church Father, when I read it in French. I try to consider the context, and lately, trying to read the psalm from a christological perspective, whether prophecy, Jesus’s own suffering on the Cross, etc.
Since this reading is done in silence, my practice of the LOTH has lots if silence in it, between the psalms and after the readings. I do the intercessions in French though.
I have a harder time having it sink in when I simply read the LOTH in French, for instance when traveling. But even if it doesn’t sink in, by being the liturgy of the Church, it is not without value to add one’s voice to the chorus of the faithful praying the LOTH.
Porthos is bang on: “it’s not all about me”, we are praying for all the faithful whether they are rejoicing or sorrowful, no matter if we share their mood or not. It’s a very important aspect of the LOTH.