hi pete, just think of it as another way to express a cause or circumstance.
e.g. line 11 of Book 1 gives a cause for what happens in line 10. Apollo did something because Agamemnon did something. You could summarise it as:
á¼ˆÏ€á½¹Î»Î»Ï‰Î½ á½¦ÏÏƒÎµ Î½Î¿á¿¦ÏƒÎ¿Î½, Î¿á½•Î½ÎµÎºÎ± á¼ˆÎ³Î±Î¼á½³Î¼Î½Ï‰Î½ á¼ Ï„á½·Î¼Î±ÏƒÎµÎ½ á¼€ÏÎ·Ï„á¿†ÏÎ±.
Note the cause is expressed by a particle Î¿á½•Î½ÎµÎºÎ± + a nominative subject á¼ˆÎ³Î±Î¼á½³Î¼Î½Ï‰Î½.
Instead of saying it that way, you can leave out the particle Î¿á½•Î½ÎµÎºÎ± and just put the subject in the genitive, and the verb in the genitive and participle form, eg
á¼ˆÏ€á½¹Î»Î»Ï‰Î½ á½¦ÏÏƒÎµ Î½Î¿á¿¦ÏƒÎ¿Î½, á¼ˆÎ³Î±Î¼á½³Î¼Î½Î¿Î½Î¿Ï‚ á¼€Ï„Î¹Î¼á½±ÏƒÎ±Î½Ï„Î¿Ï‚ á¼€ÏÎ·Ï„á¿†ÏÎ±.
Note that the subject of the genitive absolute needs to be different to the subject of the main clause. That's the point, it's an absolute, ie independent, clause. eg in line 9, you have another circumstance attached to this event, ie Apollo having been angered:
á½ƒ Î³á½°Ï Î²Î±ÏƒÎ¹Î»á¿†ÏŠ Ï‡Î¿Î»Ï‰Î¸Îµá½·Ï‚
But since Apollo is the subject of "having been angered" and of the main clause as well, you can't use gen absolute here: that's why á½ƒ ... Ï‡Î¿Î»Ï‰Î¸Îµá½·Ï‚ is nominative, not genitive. ie you could say:
á¼ˆÏ€á½¹Î»Î»Ï‰Î½, Ï‡Î¿Î»Ï‰Î¸Îµá½·Ï‚, á½¦ÏÏƒÎµ Î½Î¿á¿¦ÏƒÎ¿Î½, á¼ˆÎ³Î±Î¼á½³Î¼Î½Î¿Î½Î¿Ï‚ á¼€Ï„Î¹Î¼á½±ÏƒÎ±Î½Ï„Î¿Ï‚ á¼€ÏÎ·Ï„á¿†ÏÎ±.
This now attaches 2 circumstances to the central idea that á¼ˆÏ€á½¹Î»Î»Ï‰Î½ á½¦ÏÏƒÎµ Î½Î¿á¿¦ÏƒÎ¿Î½. Apollo is the subject of the 1st circumstance (being angered) and so is in the nom; Agamemnon the subject of the 2nd circumstance (dishonouring) and so is in the gen.
Goodwin 1897 s847 says that genitive absolutes were first used to express time, i.e. when the action in the main clause took place; later, it was used to express other circumstances, e.g. causes or conditions of the action in the main clause (as in my e.g. above). Chantraine 1963 however says that it's unclear whether the gen absolute had its origin in its temporal or causal usages, and he notes that it can be used in Homer in many ways: to express time, a condition, a cause, in a concessive use, &c (Vol 2 pages 323-324).