An enacting clause, or enacting formula, is a short phrase that introduces the main provisions of a law enacted by a legislature. It usually declares the source from which the law claims to derive its authority. In many countries, an enacting formula is not considered necessary and is simply omitted.
The simplest enacting clauses merely cite the legislature by which the law has been adopted; for example the enacting clause used in Australia since 1990 is "The Parliament of Australia enacts". Alternatively an enacting clause may invoke the ultimate sovereign. For example, California, based on the principle of popular sovereignty, has the following enacting clause: "The People of the State of California do enact as follows."
For example, Acts of the US Congress begin with words not unlike those of Acts of Parliament:
Be it enacted by the Senate and House of Representatives of the
United States of America in Congress assembled
UK statutory instruments go further, stating not only who made the order, but also which legislation provided the power to make this order. To take a recent example at random:
The Secretary of State for Education makes the following Order in exercise of the powers conferred by section 129(1) of the Education Reform Act 1988
From these, one might surmise that the purpose of these sentences is to convince the reader that the person or body issuing the legislation has the power to do so.
For most statutory instruments, the power to make them comes from Acts of Parliament, and so the specific section(s) of the those Acts are stated. For Acts of Parliament themselves, it is taken as read that the Queen, Lords and Commons acting together (or more formally, the Queen-in-Parliament) have the power to legislate, so it is sufficient to merely state that is they who have done so - and similarly for the US Congress and other legislatures.
In response to edits in the question:
Can't the hackneyed sentence ... be shortened?
Yes it could, as evidenced by the fact that not all countries use an enacting clause at all, and some of those that do have shorter, simpler ones.
Why not something less pompous?
Why should they?
One shouldn't underestimate the role of tradition in the UK constitution - indeed it's held together in part by unwritten conventions whose long use has solidified them into something like law.
As mentioned in the question, bills don't become acts until they've been assented to by a hereditary monarch, which is indicated by a clerk uttering a statement in Norman French. Compared to that, what's wrong with acts starting with a grandiose statement?
Incidentally, if you thank that's pompous, then you'll enjoy the enacting clause which appears at the top of UK Finance Acts:
Most Gracious Sovereign
WE, Your Majesty’s most dutiful and loyal subjects, the Commons of the United Kingdom in Parliament assembled, towards raising the necessary supplies to defray Your Majesty’s public expenses, and making an addition to the public revenue, have freely and voluntarily resolved to give and to grant unto Your Majesty the several duties hereinafter mentioned; and do therefore most humbly beseech Your Majesty that it may be enacted, and be it enacted by the Queen’s most Excellent Majesty, by and with the advice and consent of the Lords Spiritual and Temporal, and Commons, in this present Parliament assembled, and by the authority of the same, as follows:—
Finally, context is everything. I would strongly recommend watching a recording of the State Opening of Parliament. In addition to all the pomp and circumstance, the main event where is the Queen - wearing a priceless bejewelled crown, sitting on a golden throne under a golden canopy, watched by Peers of the Realm dressed in ermine robes - reads a speech prepared by the government, in the chamber of the House of Lords, with its golden ceiling, intricate carvings, statues and stained-glass windows. Compared to that, a few extra words at the start of an Act seems kind of modest, don't you think?