What's the point of omitting the "e", as in "sceptered" going to "scepter'd", in English poetry?

These are a few of my favorite lines of Shakespearean poetry:

Methinks I am a prophet new inspir’d,
And thus expiring do foretell of him:
His rash fierce blaze of riot cannot last,
For violent fires soon burn out themselves;
Small showers last long, but sudden storms are short;
He tires betimes that spurs too fast betimes;
With eager feeding food doth choke the feeder:
Light vanity, insatiate cormorant,
Consuming means, soon preys upon itself.

This royal throne of kings, this scepter’d isle,
This earth of majesty, this seat of Mars,
This other Eden, demi-paradise,
This fortress built by Nature for herself
Against infection and the hand of war,
This happy breed of men, this little world,
This precious stone set in the silver sea,
Which serves it in the office of a wall,
Or as a moat defensive to a house,
Against the envy of less happier lands,
This blessed plot, this earth, this realm, this England.

What I don't understand, though, is the removal of the 'e' by Shakespeare, and other poets who do this, like Tennyson. I've always read diacritics are applied by some poets to change the pronunciation, so as to keep the meter, but how on Earth is inspired pronounced differently from inspir'd, or sceptered from scepter'd? Why are they doing this?

This was to signify that the syllable was omitted. In most cases today, we don't pronounce the final syllable in many -ed endings that used to always be articulated. You can see a remnant of this in the word learned: We say that word in one syllable in the sentence

I learned a lot from him.

but pronounce it as two syllables in

He was a very learned individual.

The idea of the apostrophe was to show that the syllable was being left out so that the line would scan correctly. In the first line of your example, pronouncing the final syllable -ed *inspired* would ruin the iambic pentameter and leave the last syllable unstressed.

At one time, inspired would have been pronounced as in•spi•red, and sceptered would have been scep•ter•ed. Shakespeare, Tennyson, and other poets were just making sure their verses would scan for the reader.