Well, it's not easy to explain, but I think the TimeManager plugin does not work with CSV layers. At least not directly. Probably because the CSV layers are not editable.
First of all you will need to copy the attributes of latitude and longitude to all the rows before importing the layer in QGIS, so that for each row a point is generated. I think it's the simplest way to import the .csv file and have a spatial entity related to each of its rows. In addition, "start" and "end" fields must be well recognized with the Integer type.
The TimeManager plugin exposes, when it works, the
animation_datetime() function that returns the timestamp of the current animation. The
epoch function can converte the timestamp to Unix epoch, but in milliseconds and believing that the timestamp is in the time zone of the machine (GMT-03:00 in my case). I don't know why, but I had to recalculate it.
So, one option is to add a rule in the symbology of the CSV layer, which in my case was like this:
("start" <= epoch(animation_datetime())/1000 - 3*60*60)
("end" >= epoch(animation_datetime())/1000 - 3*60*60)
But still, the plugin needs a valid layer to work. I had created a dot shape layer previously, so the plugin worked with it. And CSV layer symbology works itself with that rule. You should create one with at least one dot that contains the start and end field for the first start and last end of the sequence. Add it to the plugin to work and disable it so it does not show up.
The other option, which seems to be much simpler, is to export the CSV layer to a dot shape layer and load it in the plugin.