It doesn't need to be activated by the words. It's activated when a person of strong spirit (and hope) wields it.
Seen in the case of Frodo when he first remembers of the Phial:
'The star-glass?' muttered Frodo, as one answering out of sleep.... 'Why yes! Why had I forgotten it? A light when all other lights go out! And now indeed light alone can help us.'
Slowly his hand went to his bosom and he held aloft the Phial of Galadriel. For a moment it glimmered, faint as a rising star struggling in heavy earthward mists, and then as its power waxed, and hope grew in Frodo's mind, it began to burn, and kindled to a silver flame..., as though Eärendil had himself come down from the high sunset paths with the last Silmaril upon his brow. The darkness receded....
Frodo gazed in wonder at this marvellous gift that he had so long carried, not guessing its full worth and potency. Seldom had he remembered it on the road..., and never had he used it for fear of its revealing light. Aiya Eärendil Elenion Ancalima! he cried, and knew not what he had spoken; for it seemed that another voice spoke through his, clear, untroubled by the foul air of the pit.
The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair
Frodo shouts the words "Aiya Eärendil Elenion Ancalima!" automatically after the Phial is lit. As Frodo become mentally stronger, the Phial becomes more powerful.
The second time it is 'activated' is when Sam wields it during his fight with Shelob:
As if his indomitable spirit had set its potency in motion, the glass blazed suddenly like a white torch in his hand. It flamed like a star that leaping from the firmament sears the dark air with intolerable light.
The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair
For your follow-up question: Frodo doesn't drop the Phial at all in the book. He hands it over to Sam, who keeps it with him.
‘That would not help us now,’ said Frodo. ‘Come! Let us see what Sting can do. It is an elven-blade. There were webs of horror in the dark ravines of Beleriand where it was forged. But you must be the guard and hold back the eyes. Here, take the star-glass. Do not be afraid. Hold it up and watch!’
[...]
‘Orcs!’ he muttered. ‘We’ll never rush it like this. There’s Orcs about, and worse than Orcs.’ Then returning quickly to his long habit of secrecy, he closed his hand about the precious Phial which he still bore. Red with his own living blood his hand shone for a moment, and then he thrust the revealing light deep into a pocket near his breast and drew his elven-cloak about him. Now he tried to quicken his pace. His master was gaining on him; already he was some twenty strides ahead, flitting on like a shadow; soon he would be lost to sight in that grey world.
The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair
Sam then stows the Phial into a pocket, but he never drops it.
In the film adaptation of The Return of the King however, it is shown that the Phial is activated only after Frodo chants the words.

Non-canonical of course, but it made a pretty good scene.